A quadratic-time coloring algorithm for graphs with large maximum degree

A quadratic-time coloring algorithm for graphs with large maxim um degree F eng Liu ∗ Sh uang Sun † Y an W ang ‡ Sc ho ol of Mathematics Sciences, Shanghai Jiao T ong Univ ersity , 800 Dongch uan Road, Shanghai, 200240, China Marc h 18, 2026 Abstract Graph coloring is a cen tral problem in graph theory and is NP-hard for general graphs. Motiv ated by the Boro din–K osto chk a conjecture, we study the algorithmic problem of coloring graphs with large maximum degree and no clique of size ∆ . W e giv e a quadratic-time coloring algorithm that constructs a (∆ − 1) -coloring for such graphs. W e also prov e that every graph G with maximum degree ∆ ≥ 7 . 3 × 10 9 and clique num b er ω ( G ) < ∆ s atisfies χ ( G ) ≤ ∆ − 1 . This impro ves a longstanding result of Reed. Keyw ords: Graph coloring; Borodin–Kostochk a conjecture; maximum degree; coloring algo- rithm. AMS Sub ject Classification: 05C15, 05C85. 1 In tro duction All graphs considered in this pap er are finite and simple. Let G = ( V , E ) b e a graph. F or a p ositiv e in teger k , a k -coloring of a graph G is a mapping f : V → { 1 , 2 , . . . , k } suc h that f ( u )  = f ( v ) whenev er u and v are adjacen t in G . A graph is k -colorable if it admits a k -coloring. The chromatic n umber of G , denoted b y χ ( G ) , is the minimum n umber k for whic h G is k -colorable. Clearly , χ ( G ) ≤ ∆( G ) + 1 . In 1941, Brooks [ 3 ] pro v ed the first non trivial result ab out coloring graphs with at most ∆( G ) colors: Theorem 1.1 (Bro oks [ 3 ]) . If G is a c onne cte d gr aph with ∆( G ) ≥ 3 , then χ ( G ) ≤ max { ∆( G ) , ω ( G ) } . In 1977, Boro din and K osto c hk a [ 2 ] prop osed the following conjecture. Conjecture 1.2 (Borodin-Kostochk a [ 2 ]) . If G is a c onne cte d gr aph with ∆( G ) ≥ 9 , then χ ( G ) ≤ max { ∆( G ) − 1 , ω ( G ) } . By Theorem 1.1 , if ω ( G ) ≥ ∆( G ) , then Conjecture 1.2 holds immediately . Therefore, to pro ve the conjecture, it suffices to pro ve that for a graph G , if ∆( G ) ≥ 9 and ω ( G ) ≤ ∆( G ) − 1 , then ∗ Email: liufeng0609@126.com. † Email: c ho colatesun@sjtu.edu.cn. ‡ Email: y an.w@sjtu.edu.cn (corresp onding author). 1 χ ( G ) ≤ ∆( G ) − 1 . Cranston and Rab ern [ 10 ] sho wed that Conjecture 1.2 cannot b e strengthened b y replacing the condition ∆( G ) ≥ 9 with ∆( G ) ≥ 8 or by replacing the clique condition with ω ( G ) ≤ ∆( G ) − 2 . By Theorem 1.1 , each graph G with χ ( G ) > ∆( G ) ≥ 9 con tains a (∆( G ) + 1) - clique. So, Conjecture 1.2 is equiv alen t to the statement that each graph G with χ ( G ) = ∆( G ) ≥ 9 con tains a ∆( G ) -clique. In 1999, Reed [ 18 ] presen ted a partial result to wards Conjecture 1.2 by showing that the conjec- ture is true for all graphs G with ∆( G ) ≥ 10 14 . Muc h w ork has also b een dev oted to proving Conjecture 1.2 for sp ecial graph classes. In 2013, Cranston and Rab ern [ 9 ] prov ed the conjecture for claw-free graphs. Later, Cranston, Lafay ette and Rab ern [ 8 ] extended this result to { P 5 , gem } -free graphs. Gupta and Pradhan [ 14 ] established the conjecture for { P 5 , C 4 } -free graphs. Lan, Liu and Zhou [ 16 ] prov ed it for { P 2 ∪ P 3 , C 4 } -free graphs, and Chen, Lan and Zhou [ 7 ] further pro ved it for { P 2 ∪ P 3 , house } -free graphs. More recently , Chen, Lan, Lin and Zhou [ 5 ] pro ved that Conjecture 1.2 holds for odd-hole-free graphs. They subsequen tly strengthened this result by sho wing that a connected odd-hole-free graph G satisfies χ ( G ) ≤ max { ∆( G ) − 1 , ω ( G ) } if and only if G  ∼ = C 7 , and in general every o dd-hole-free graph satisfies χ ( G ) ≤ max { 4 , ∆( G ) − 1 , ω ( G ) } . There are also further extensions to other restricted graph classes. In particular, W u and W u [ 19 ] prov ed Conjecture 1.2 for ( P 6 , apple , torc h) -free graphs, thereb y generalizing the theorem for { P 5 , C 4 } -free graphs due to Gupta and Pradhan [ 14 ]. Another related line of researc h studies structural w eakenings of Conjecture 1.2 . Since the conjecture predicts a clique of size ∆( G ) whenev er χ ( G ) = ∆( G ) ≥ 9 , it is natural to ask whether one can still force some strong extremal structure under the same h yp othesis, ev en if one cannot pro ve the existence of a ∆( G ) -clique. In this direction, recent w ork has sho wn that graphs with χ ( G ) = ∆( G ) often contain either a large clique or a high o dd hole. In particular, Galindo, McDonald and Shan [ 13 ] prov ed that if G is connected, χ ( G ) = ∆( G ) , and G  = C 7 , then G con tains either K ∆( G ) or a high o dd hole. Existing pro ofs of exact Borodin-Kostochk a-t yp e results on hereditary graph classes usually re- duce to the case ∆ = 9 and then analyze a minimal vertex-critical coun terexample. This reduction go es bac k to Catlin[ 4 ] and K osto chk a[ 15 ] and has b ecome a standard to ol in the area. One then com bines color criticality , local recoloring, and the forbidden induced subgraph assumptions to de- riv e a contradiction. By contrast, Reed’s theorem works in the unrestricted setting and relies on probabilistic metho ds rather than a finite forbidden-configuration analysis. These t wo lines of work indicate tw o rather different approaches to w ards the conjecture: exact results for sp ecific hereditary classes, and asymptotic results for general graphs. In this paper, w e work in the second direction. Moreo ver, we aim to provide a constructive coloring algorithm for graphs with large maxim um degree. W e first state our algorithmic result. Theorem 1.3. Ther e is a quadr atic-time (∆( G ) − 1) -c oloring algorithm for gr aphs with maximum de gr e e ∆( G ) at le ast 7 . 4 × 10 9 . The algorithmic result is supp orted b y the following existence theorem. Theorem 1.4. Every gr aph G with ∆( G ) ≥ 7 . 4 × 10 9 satisfies χ ( G ) ≤ max { ∆( G ) − 1 , ω ( G ) } . In particular, every graph G with ∆( G ) ≥ 7 . 4 × 10 9 and ω ( G ) < ∆( G ) is (∆( G ) − 1) -colorable. 2 Our pro of com bines a structural analysis of the graph with the Moser–T ardos algorithmic lo cal lemma, yielding b oth an existence result and an explicit quadratic-time recoloring pro cedure. Our pro of follows the general outline of Reed’s argumen t, but w e reorganize it in a wa y that can b e made constructiv e. W e first deriv e a structural decomp osition for a minimal coun terexam- ple, isolating large cliques and near-cliques. W e then use a probabilistic coloring scheme, together with Azuma’s inequalit y and the Lov ász Lo cal Lemma, to obtain a partial coloring satisfying the h yp otheses of a deterministic extension lemma. Finally , we apply the Moser–T ardos algorithmic lo cal lemma to con v ert the existence pro of into an explicit recoloring pro cedure, and w e refine the implemen tation to obtain a quadratic-time coloring algorithm. The remainder of this paper is organized as follo ws. In Section 2 , we presen t a structural c haracterization of graphs in this class. In Section 3 , we derive a coloring scheme b y applying the structural theorem together with the Lo v ász Lo cal Lemma. In Section 4 , w e describe a quadratic- time coloring algorithm. 2 Structural decomp osition In this section, we pro vide a structural c haracterization of the minimal coun terexample, which facilitates the subsequen t presentation of a coloring scheme and algorithm. Lemma 2.1 (Reed [ 18 ]) . If G is a minimal c ounter example to the Bor o din-Kosto chka c onje ctur e of maximum de gr e e ∆ , then its ∆ − 1 cliques ar e disjoint. Lemma 2.2 (Reed [ 18 ]) . If G is a minimal c ounter-example to The or em 1.4 of maximum de gr e e ∆ , and K is a ∆ − 1 clique of G then no vertex of G − K is adjac ent to mor e than four vertic es of K . F urthermor e, at most four vertic es of K have de gr e e ∆ − 1 . Lemma 2.3. If G is a minimal c ounter example to the Bor o din-Kosto chka c onje ctur e of maximum de gr e e ∆ , then no ∆ − 1 cliques interse ct ∆ − 2 cliques in ∆ − 3 vertic es. Pr o of. Supp ose there exists ∆ − 1 clique which intersect with a ∆ − 2 clique C in ∆ − 3 vertices. Let x b e the v ertex of ∆ − 2 clique and y 1 , y 2 b e the tw o vertices in a ∆ − 1 clique. Note that x is not adjacen t to y 1 , y 2 and y 1 y 2 ∈ E ( G ) . If no vertex of C has any neighbor in G − C − x − y 1 − y 2 . As abov e, we can extend this to a coloring of G − C in whic h x and y 1 ha ve the same color. Now, since each vertex of C has at most ∆ − 1 neigh b ors and adjacent to b oth x and y 1 , w e can greedily complete this coloring to a ∆ − 1 coloring of G . Let H = G − C − x − y 1 − y 2 . So, we assume there is a vertex z of G − C − x − y adjacen t to a vertex in C . If z is complete to C , then it misses x, y 1 and y 2 or it is a larger clique. In this case, we ∆ − 1 color H − z and extend this coloring to a ∆ − 1 coloring of G − C by coloring all of x, y 1 , z with the same color, one which app ears on none of the at most nine vertices of H adjacent to an elemen t of this triple. W e can now greedily extend this coloring to a ∆ − 1 coloring of G b ecause C is complete to { x, y 1 , z } . Th us, we can assume that z has an non-neighbor w in C . Note that w is complete to C + x + y 1 + y 2 − w and hence has at most one neigh b or in H . If z has three or more neighbors in C , then w e will extend a ∆ − 1 coloring of H − z to a ∆ − 1 coloring of G . W e first color z and w with some color i which appears neither on any of the at most ∆ − 3 neigh b ors of z in H nor on the neigh b or of 3 w in H . W e next color b oth x and y 1 with a color j different from i whic h app ears on none of the at most five vertices of H adjacent to at least one vertex of this pair. Then, w e color y 2 with a color differen t from i, j which app ears on none of the at most t wo vertices of H adjacen t to y 2 . Finally , w e color the vertices of C − w saving some neighbor v of z to color last. When w e come to color a v ertex u of C − v , there will b e one uncolored neighbor v of u , and there will b e a pair of neighbors of v with the same color. Th us, we can greedily extend our coloring of G − C to a ∆ − 1 coloring of G − v . Finally , v is adjacent to w, x, y 1 , z but we used only tw o colors on these four v ertices so w e can actually extend our coloring to a ∆ − 1 coloring of G , a contradiction. Th us, w e can assume that z has at most t wo vertices of C . W e can also assume that every non-neigh b or w of z has a neighbor in H . Otherwise, we can ∆ − 1 color H , color w with the same color as z and extend this coloring to a coloring of G as w e did in the last paragraph. Our next step is to show that there is some vertex w of C not adjacent to z suc h that the graph H z w obtained from H by adding an edge from z to the neigh b or of w in H con tains no ∆ clique. T o this end, note that if H z w con tains a ∆ clique then by Lemma 2.1 there exists tw o ∆ − 1 cilques in terest, a con tradiction. It follo ws that we can c ho ose some non-neighbor w of z in C such that H z w con tains no ∆ clique. No w, w e note that ∆( H z w ) ⩽ ∆ as z has a neighbor in C and w has only one neighbor in H . Th us, by the minimality of G , we can ∆ − 1 color H z w . This yields a ∆ − 1 coloring of H + w in whic h z and w ha ve the same color. As abov e, we can extend this to a ∆ − 1 coloring of G if w e first color x and y with the same color and sav e a neigh b or v of z to color last. This con tradiction completes the pro of of Lemma 2.4 . Based on Lemma 2.3 , using the same proof tec hnique w e can obtain the following lemma. The details are en tirely analogous and are omitted here. Lemma 2.4. If G is a minimal c ounter example to the Bor o din-Kosto chka c onje ctur e of maximum de gr e e ∆ , then no ∆ − 2 cliques interse ct in ∆ − 3 vertic es. Lemma 2.5. If G is a minimal c ounter example to The or em 1.4 of maximum de gr e e ∆ , and K is a ∆ − 2 clique of G then no vertex of G − K is adjac ent to mor e than 5 vertic es of K . F urthermor e, at most 5 vertic es of K have de gr e e ∆ − 1 . Pr o of. Supp ose there is a ∆ − 2 clique K and a v ertex not in K which has at least fiv e neigh b ors in K . Then, w e let x b e a v ertex of G − K which has the maximum num b er of neighbors in K , b y Lemma 2.3 , 2.4 w e know that this is at most ∆ − 3 . Let y b e a non-neighbor of x in K , let C = K − y and let D be the neighborho o d of x in C . If there is no v ertex of D adjacent to any v ertex of G − C − x − y , then eac h vertex in D has degree ∆ − 2 , a con tradiction. Thus, we can assume there is a vertex z in H = G − C − x − y adjacen t to a vertex in D . If z has five or more neigh b ors in D , then we will extend a ∆ − 1 coloring of H − z to a ∆ − 1 coloring of G . T o do so, there exists some non-neighbor w of z in C other than y , otherwise contrary to Lemma 2.3 or 2.4 . W e first color z and w with some color i which app ears neither on an y the at most ∆ − 5 neighbors of z in H nor on the at most 3 neighbors of w in H . W e next color b oth x and y with a color differen t from i which app ears on none of the at most ∆ − 2 vertices of H adjacen t to at least one vertex of this pair. Next, we color the v ertices of C − D , all of whic h ha ve t wo uncolored 4 neigh b ors in D . Finally , we color the v ertices of D sa ving some neighbor v of z to color last. When w e come to color a vertex u of D − v , there will b e one uncolored neighbor of u : v and there will b e a pair of neighbors of u with the same color: { x, y } . Thus, w e can greedily extend our coloring of G − D to a ∆ − 1 coloring of G − v . Finally , v is adjacent to w , x, y , z but we used only tw o colors on these four vertices so we can actually extend our coloring to a ∆ − 1 coloring of G , a con tradiction. Th us, w e can assume that z has at most four neighbors in D , and hence misses a vertex w in D . No w, if w has exactly a neighbor u in H , we set H ′ = H + zu . Otherwise, w e set H ′ = H . In either case, by Lemma 2.1 , we know H ′ con tains no ∆ clique. F urther, ∆( H ′ ) ⩽ ∆ as z and u hav e neigh b ors in C . Th us, by the minimalit y of G , w e can ∆ − 1 color H ′ . This yields a ∆ − 1 coloring of H + w in whic h z and w ha v e the same color. As abov e, we can extend this to a ∆ − 1 coloring of G if w e first color x and y with the same color and sav e a neighbor v of z in D to color last. No w, if w has tw o neighbors u 1 , u 2 in H , we set H ′ = H + z u 1 + z u 2 . Otherwise, we set H ′ = H . In either case, b y Lemma 2.1 , we know H + z u 1 and H + z u 2 con tain no ∆ clique. If H ′ con tains ∆ clique, w e ha ve u 1 u 2 ∈ E ( G ) , and this con trary to Lemma 2.3 . Then we can also color G with ∆ − 1 colors like b efore. No w, supp ose there is some ∆ − 2 clique C in G containing a set S of six v ertices each with degree ∆ − 1 . Let v be a vertex of S , and let z be the neighbour of v outside C . Since z has at most four neighbours in C , there is at least one non-neighbour w of z in S . W e can mimic the pro of ab o ve to find a coloring of G − C + w in which z and w ha ve the same color. W e can complete the coloring if w e color v last and some other vertex of S second last. This prov es Lemma 2.5 Lemma 2.6. If G is a minimal c ounter example to The or em 1.4 and H is a sub gr aph of G with at most ∆ + c, c ≥ 6 vertic es such that every vertex of H has at le ast 4 / 5∆ neighb ors in H then H is either a clique or c onsists of a clique C H with less than ∆ − 1 vertic es and a vertex v H . Pr o of. Consider a minimal counterexample G to Theorem 1.4 , and a set H with at most ∆ vertices whic h induces a graph of minim um degree 4 / 5∆ . By our degree condition, for eac h pair { x, y } of v ertices of H , there m ust b e at least 2∆ / 5 vertices in the set S x,y = N ( x ) ∩ N ( y ) ∩ H . Th us, if H has 3 disjoin t pairs of nonadjacent v ertices { ( x 1 , y 1 ) , . . . , ( x 3 , y 3 ) } , then there must b e at least ∆ / 11 > 100 vertices which are in tw o of the S x i ,y i . In particular this implies that H contains t wo vertices a and b such that N ( a ) ∩ N ( b ) ∩ H con tains t w o disjoint pairs of nonadjacen t vertices, call them ( x, y ) and ( v , w ) . By the minimalit y of G , we can ∆ − 1 color G − H . W e can extend this to a coloring of G − H + x + y + w + v in which ( x, y ) and ( w , v ) are pairs of v ertices with the same color, b ecause of our degree condition on H . Set S = a + b + ( N ( a ) ∩ N ( b ) ∩ H ) − v − w − x − y . W e next extend our coloring to a ∆ − 1 coloring of G − S b y first coloring those v ertices of G − S whic h ha ve at most one neighbor in S and then coloring the set T of v ertices of G − S whic h ha ve at least t wo neigh b ors in S . W e note that our degree condition ensures that | S | ⩾ ∆ / 7 − c . If | S | ≤ 3∆ / 5 − 1 then eac h v ertex of S m ust b e adjacen t to at least 4 / 5∆ + 1 − | S | vertices of H − S and hence there are at least | S | (4 / 5∆ + 1 − | S | ) edges from S to H − S . Since there are at most | T || S | + | S | edges from S to H − S , ( | T || S | + | S | ) ≥ | S | (4 / 5∆ + 1 − | S | ) . In this case, | S ∪ T | ⩾ 3∆ / 5 . Th us, in either case, | S ∪ T | ⩾ 3∆ / 5 .. So, each vertex of H − S − T has at least tw o neighbors in S ∪ T and there is no problem coloring them. Each vertex of T has tw o neigh b ors in S , so w e can indeed extend our ∆ − 1 coloring to G − S . No w, w e can color all of S − a − b , because eac h vertex in this set is adjacen t to both a and b . Finally , we can color a and b because b oth their neighborho o ds con tain 5 the four v ertices v , w , x, y on which we used only tw o colors. This is a con tradiction. So, we can assume that H has no 3 disjoin t pairs of non-adjacent vertices. Hence by considering the clique obtained b y deleting a maximum family of pairs of disjoint pairs of non-adjacent vertices of H , we know that if we let C H b e a maximum clique of H then H − C H has at most 4 v ertices. If H − C H has at least t wo vertices then b y the maximality of C H , there are either tw o pairs of nonadjacen t vertices in H or there is a stable set of size three in H . That is, there are four vertices of H whic h p ermit a tw o coloring, and more strongly , we can choose t wo of these vertices in C H . In either case, w e can color G − H with ∆ − 1 colors by the minimality of G , extend this b y coloring some set X of four v ertices of H including tw o in C H with only tw o colors, then color H − C H , and finally color C H , sa ving t wo of the at least ∆ / 5 vertices which is adjacen t to all vertices of X to color last. This will yield a ∆ − 1 coloring of G , a con tradiction. So, w e see that H do es indeed consist of a clique C H and a v ertex v H . F urthermore, if H is not a clique then b y Lemma 2.1 , C H has at most ∆ − 2 vertices . Corollary 2.7. If two elements C 1 and C 2 of C with | C 1 | ⩽ | C 2 | interse ct, then | C 1 − C 2 | ⩽ 1 . A nd no element of C interse cts two other elements of C . This result implies that w e can partition V ( G ) into sets S 1 , . . . , S ℓ , where L = V ( G ) \ [ { S i | S i ∈ S } , so that eac h S i is either: • a clique C i ∈ C , or • a set consisting of a clique C i ∈ C together with a vertex u i ∈ V ( G ) \ C i that has at least 4 5 | C i | neigh b ours in C i . In the second case, w e say S i is a ne ar-clique . W e shall need the following results concerning this partition. Lemma 2.8 (Reed [ 18 ]) . If v is a vertex in some C i such that | C i | = ∆ − p then ther e is a most one neighb or of v outside C i which has mor e than p + 3 neighb ors in C i . F urthermor e, if v has de gr e e ∆ − 1 ther e ar e no such neighb ors. Lemma 2.9. F or any C i , if | C i | = ∆ − p then we c an find at le ast ∆ / 9 disjoint triples e ach of which c onsists of a vertex v of C i and two neighb ors of v outside of C i b oth of which have at most p + 3 neighb ors in C i . Pr o of. T ake a maximal such set of triples, supp ose it has k elemen ts. Let S be the set of 2 k v ertices outside C i con tained in one of these triples. Let T b e the set of k v ertices of C i con tained in some triple. Since G has minim um degree at least ∆ − 1 , the maximalit y of our triple set and Lemma 2.8 imply that each vertex of C i − T has at least p − 1 neighbors in S . Thus there are at least ( p − 1)(∆ − k ) edges b et ween S and C i − T . On the other hand there are at most 2 k ( p + 3) such edges. So, if p is at least 2 , then the desired result holds. If p = 1 , then the result holds b y Lemma 2.2 , b ecause all but four of the v ertices of C i ha ve at least t w o neigh b ors in G − C i eac h of whic h has degree at most four in C i . If p = 2 , then the result holds by Lemma 2.5 , b ecause all but fiv e of the v ertices of C i ha ve at least t wo neighbors of G − C i eac h of whic h has degree at most four in C i . Hence we can actually find a disjoin t set of (∆ − 4) / 9 disjoin t triples. 6 Lemma 2.10. If v is a vertex whose neighb orho o d c ontains fewer than ∆ 2 / 50 − ∆ / 10 p airs of non-adjac ent vertic es then it is in some S i . Pr o of. If v is not in an y S i then by Lemma 2.6 , there is some vertex v 0 in N ( v ) which is adjacent to less than 4 / 5∆ − 1 v ertices of N ( v ) . More strongly , there is a sequence v 0 , . . . , v 1 / 5∆ of v ertices of G suc h that v i is adjacen t to less than 4 / 5∆ v ertices of N ( v ) − { v j | j < i } . The result follows. 3 A prop er coloring In this section, w e will use the Lov ász Lo cal Lemma to give a feasible coloring scheme. Lemma 3.1 (Reed [ 18 ]) . Any p artial ∆ − 1 c oloring of G satisfying the thr e e fol lowing c onditions c an b e extende d to a ∆ − 1 c oloring of G . (i) for every vertex v ∈ L ther e ar e at le ast 2 c olors app e aring twic e in the neighb orho o d of v , (ii) for e ach ne ar clique S i , ther e ar e two unc olor e d neighb ors of v i in C i , and (iii) for every C i , ther e ar e two unc olor e d vertic es w i and x i of C i whose neighb orho o ds c ontain two r ep e ate d c olors. Theorem 3.2 (Lo v ász Lo cal Lemma [ 12 ]) . L et X b e a set of events such that e ach X ∈ X satisfies: ( i ) P ( X ) ≤ p and ( ii ) X is mutual ly indep endent of a set of al l but at most d other events of X . If ep ( d + 1) ≤ 1 then with p ositive pr ob ability, none of the events in X o c cur. Theorem 3.3 (Azuma’s Inequality [ 1 ]) . L et X b e a r andom variable determine d by n trials tr 1 , ..., tr n . If for e ach i and for any two p ossible se quenc es of outc omes t 1 , . . . , t i − 1 , t i and t 1 , . . . , t i − 1 , t i ′ the fol lowing holds: | E ( X | tr 1 = t 1 , ..., tr i = t i ) − E ( X | tr 1 = t 1 , ..., tr i = t i ′ ) | ≤ a i then P ( | X − E ( X ) | > t ) ≤ 2 e − t 2 / (2Σ a 2 i ) . 3.1 Ensuring t wo rep eated colors in N ( v ) via Azuma and the Lo cal Lemma Fix a v ertex v ∈ L . As in [ 18 ], w e exp ose a uniformly random (∆ − 1) -coloring of V ( G ) and then uncolor every vertex in volv ed in a conflict. Let U v b e the set of unordered pairs { u, w } of non- adjac ent v ertices in N ( v ) such that, in the resulting partial coloring, u and w retain the same color and this color appears no where else in N ( v ) . Let W v b e the set of colors used on the pairs in U v , and define Z v := | U v | = | W v | . Clearly , Z v is a low er b ound on the num b er of colors that app ear at least twice in N ( v ) . Set the bad ev ent A v := { Z v < 2 } . 7 Let M v denote the n umber of non-adjacent pairs in N ( v ) . By Lemma 2.10 , M v ≥ ∆ 2 50 − ∆ 10 . Fix a non-adjacent pair { u, w } ⊆ N ( v ) . The probabilit y that u and w receive the same color is 1 / (∆ − 1) . Condition on this even t and let c b e their common color. A sufficien t condition for { u, w } to contribute to Z v is that no other v ertex in N ( u ) ∪ N ( w ) ∪ N ( v ) receiv es color c . Since | N ( u ) ∪ N ( w ) ∪ N ( v ) | − 3 ≤ 3∆ − 3 , the probability of this sufficien t condition is at least  1 − 1 ∆ − 1  3∆ − 3 . F or ∆ ≥ 3 we hav e x := 1 / (∆ − 1) ≤ 1 / 2 , hence ln(1 − x ) ≥ − x − x 2 . Therefore  1 − 1 ∆ − 1  3∆ − 3 = exp  (3∆ − 3) ln  1 − 1 ∆ − 1  ≥ exp  − (3∆ − 3)  1 ∆ − 1 + 1 (∆ − 1) 2  = exp  − 3 − 3 ∆ − 1  . (1) Consequen tly , P  { u, w } contributes to Z v  ≥ 1 ∆ − 1 exp  − 3 − 3 ∆ − 1  . By linearit y of exp ectation, µ := E ( Z v ) ≥  ∆ 2 50 − ∆ 10  1 ∆ − 1 exp  − 3 − 3 ∆ − 1  . (2) W e no w prov e a tail b ound on Z v using Azuma’s inequality . As in [ 18 ], we exp ose the random colors in a suitable order. Order the vertices of G as w 1 , . . . , w n so that the neighbors of v form a suffix of this order, and let w s b e the last vertex of V ( G ) \ N ( v ) . Let F i b e the σ -algebra generated b y the colors of w 1 , . . . , w i , and define the Do ob martingale X i := E  Z v | F i  , i = 0 , 1 , . . . , n, so that X 0 = µ and X n = Z v . Let c i := sup ω ,ω ′   X i ( ω ) − X i − 1 ( ω ′ )   , where the supremum is tak en ov er outcomes consisten t with F i − 1 and differing only in the color c hoice of w i . First consider indices i > s , i.e. w i ∈ N ( v ) . Changing the color of a single vertex can change Z v b y at most 2 , hence c i ≤ 2 for i > s . Since | N ( v ) | ≤ ∆ , we hav e n X i = s +1 c 2 i ≤ ∆ · 2 2 = 4∆ . 8 No w assume i ≤ s , so w i / ∈ N ( v ) . Let D i := N ( w i ) ∩ N ( v ) and d i := | D i | . Changing the color of w i from j to k can only affect membership of colors j and k in the set W v , and hence can cha nge Z v b y at most 2 . Moreo ver, for Z v to c hange due to the color j , it is necessary that j forms a unique rep eated color in N ( v ) , i.e. j appears exactly twic e in N ( v ) and one of the t wo vertices with color j lies in D i . Because N ( v ) is a suffix and i ≤ s , conditional on F i − 1 the colors on N ( v ) remain indep enden t and uniform o ver { 1 , . . . , ∆ − 1 } . Fix x ∈ D i . Conditional on col( x ) = j , the probability that j app ears exactly once in N ( v ) \ { x } equals  ∆ − 1 1  1 ∆ − 1  1 − 1 ∆ − 1  ∆ − 2 =  1 − 1 ∆ − 1  ∆ − 2 ≤ e − 1 . Th us, for fixed j , P  x is one of the t wo o ccurrences of color j in N ( v ) | F i − 1  ≤ 1 ∆ − 1 · e − 1 . T aking a union bound o ver x ∈ D i sho ws that the probabilit y that j is a unique rep eated color in N ( v ) with one o ccurrence in D i is at most d i e − 1 / (∆ − 1) . The same b ound holds for k . Therefore, the probabilit y that changing w i ’s color from j to k affects Z v is at most 2 e − 1 d i ∆ − 1 , and since | Z v | c hanges by at most 2 we may take c i ≤ 2 · 2 e − 1 d i ∆ − 1 = 4 e − 1 d i ∆ − 1 . Hence s X i =1 c 2 i ≤ 16 e 2 (∆ − 1) 2 s X i =1 d 2 i . Clearly P s i =1 d i ≤ ∆ 2 (eac h of the ∆ vertices in N ( v ) has at most ∆ neigh b ors outside N ( v ) ), and max i d i ≤ ∆ . Th us P s i =1 d 2 i ≤ (max i d i ) P s i =1 d i ≤ ∆ 3 , giving s X i =1 c 2 i ≤ 16 e 2 (∆ − 1) 2 ∆ 3 ≤ 16 e 2 ∆ . Com bining the tw o parts, we obtain n X i =1 c 2 i ≤  4 + 16 e 2  ∆ . (3) Applying Azuma’s inequalit y to the martingale ( X i ) n i =0 with a = µ − 2 , w e deduce P ( A v ) = P ( Z v < 2) ≤ 2 exp − ( µ − 2) 2 2  4 + 16 e 2  ∆ ! . (4) Finally , we apply the Lo v ász Lo cal Lemma. The random v ariable Z v dep ends only on colors within distance at most 2 from v . Therefore, A v is indep endent of A u whenev er the distance b etw een u and v is at least 5 . The num b er of vertices within distance 4 of any fixed vertex is at most 1 + ∆ + ∆(∆ − 1) + ∆(∆ − 1) 2 + ∆(∆ − 1) 3 ≤ ∆ 4 + 1 . Hence each even t A v is mutually indep endent of all but at most d := ∆ 4 + 1 other ev ents in the family . In particular, if ∆ ≥ 1 . 055 × 10 9 , then e p ( d + 1) < 1 (with p = max v ∈ L P ( A v ) ), and so with p ositiv e probability none of the even ts A v o ccur. Consequen tly , Z v ≥ 2 for every v ∈ L. 9 3.2 Bounding P ( E i ) for near-cliques W e note that each E i dep ends only on the colors of the vertices in S i and within distance one of S i . Similarly , eac h F i dep ends only on the colors of the vertices in S i and within distance tw o of S i . It follo ws that eac h even t in E is indep endent of a set of all but at most 3∆ 5 other ev en ts. Thus, it suffices to sho w that each even t in E holds with probability at most ∆ − 6 . W e hav e already established this b ound for the even ts A v . No w fix a near clique S i with sp ecial v ertex v i and clique part C i . Choose a set R i ⊆ N ( v i ) ∩ C i with | R i | = 4 5 ∆ . Let N i b e the num b er of distinct colors app earing on R i in the initial random coloring. If E i holds, then there are fewer than t w o uncolored neigh b ors of v i in C i after uncoloring conflicts. Since R i induces a clique, ev ery color that app ears at least twice on R i creates conflicts, and hence all v ertices of R i with that color are uncolored. Consequen tly , if E i holds then at most one v ertex of R i is uncolored, and therefore N i ≥ | R i | − 2 = 4 5 ∆ − 2 . (5) On the other hand, b y linearity of exp ectation, E ( N i ) = (∆ − 1) 1 −  1 − 1 ∆ − 1  | R i | ! < (∆ − 1)  1 − e −| R i | / (∆ − 1)  ≤ (∆ − 1)  1 − e − 4 / 5  , where w e used (1 − 1 /q ) m > e − m/q with q = ∆ − 1 . Set t :=  4 5 ∆ − 2  − E ( N i ) . Then for all ∆ ≥ 3 w e hav e t ≥  e − 4 / 5 − 1 5  ∆ −  1 + 12 5 e − 4 / 5  . Moreo ver, N i dep ends only on the color c hoices on the | R i | = 4 5 ∆ vertices of R i , and changing the color of a single v ertex can change N i b y at most 1 . Therefore Azuma’s inequality yields P  | N i − E ( N i ) | > t  ≤ exp  − t 2 2 | R i |  = exp − t 2 8 5 ∆ ! . By ( 5 ), this implies P ( E i ) ≤ P  N i ≥ 4 5 ∆ − 2  ≤ exp − t 2 8 5 ∆ ! . In particular, for ∆ sufficien tly large, the righ t-hand side is at most ∆ − 2 , and hence each E i o ccurs with probabilit y less than ∆ − 2 . 3.3 Estimating F i and a lo wer b ound for ∆ T o compute the probability b ound on F i , w e consider a family T i of |T i | = k ∆ pairwise v ertex-disjoint triples, where k ∈ (0 , 1 / 9] , 10 whose existence is guaran teed b y Lemma 2.9 . Let T i denote the union of the vertex sets of the triples in T i . W e define M i to b e the num b er of triples in T i for which the v ertex in C i is uncolored, b oth of the other v ertices are colored with colors whic h are also used to color vertices of C i , and no vertex of the triple is assigned a color that is assigned to an y vertex in any of the other triples. (The last condition is imp osed to ensure that changing the color of a single v ertex can affect the v alue of M i b y at most 2 .) W e b egin b y estimating E ( M i ) . W e note that M i coun ts the n umber of triples ( a, b, c ) ∈ T i with c ∈ C i suc h that there exist colors j, k , ℓ and v ertices x, y , z with x ∈ C i \  T i ∪ N ( a )  , y ∈ C i \  T i ∪ N ( b )  , z ∈ N ( c ) \ T i , satisfying the follo wing three conditions: ( i ) the color j is assigned to b oth a and x , and to none of the remaining vertices of T i ∪ N ( a ) ∪ N ( x ) ; ( ii ) the color k is assigned to b oth b and y , and to none of the remaining v ertices of T i ∪ N ( b ) ∪ N ( y ) ; ( iii ) the color ℓ is assigned to b oth z and c , and to none of the remaining vertices of T i . T o b egin, fix a triple ( a, b, c ) ∈ T i . F or distinct colors j , k , ℓ and vertices x, y , z , let A j,k,ℓ,x,y ,z b e the ev ent that (1), (2) and (3) hold. As in [ 18 ], we hav e P ( A j,k,ℓ,x,y ,z ) ≥ (∆ − 1) − 6 e − 5 , and the ev ents A j,k,ℓ,x,y ,z corresp onding to differen t index sets are pairwise disjoin t. Moreo ver, b y Lemma 2.9 we hav e at least  3 / 5 − k  ∆ c hoices for each of x and y , and at least (1 − 3 k )∆ choices for z . Since there are (∆ − 1)(∆ − 2)(∆ − 3) c hoices for distinct j, k , ℓ , we obtain P  ∃ ( j, k , ℓ, x, y , z ) suc h that (1)–(3) hold  ≥ e − 5  3 5 − k  2 (1 − 3 k ) · (∆ − 2)(∆ − 3)∆ 3 (∆ − 1) 5 . In particular, for ∆ ≥ 10 9 the last factor is at least 1 − ϵ, ϵ < 10 − 17 , and hence P  the triple ( a, b, c ) is coun ted by M i  ≥ (1 − ϵ ) e − 5  3 5 − k  2 (1 − 3 k ) . Since |T i | = k ∆ , b y linearity of exp ectation w e conclude that µ := E ( M i ) ≥ (1 − ϵ ) k  3 5 − k  2 (1 − 3 k ) ∆ e − 5 . Lemma 3.4. L et T ⟩ b e a family of p airwise vertex-disjoint triples and assume that |T ⟩ | = k ∆ for some c onstant k ∈ (0 , 1) . L et T i b e the union of the vertex sets of the triples in T ⟩ , so that | T i | = 3 k ∆ . Exp ose the r andom c olors in two phases: first c olor the vertic es of V ( G ) \ T i and then the vertic es of T i . L et ( F j ) b e the r esulting filtr ation and set X j := E ( M i | F j ) , wher e M i is define d as in the pr o of of the b ound on P ( F i ) . Then the martingale incr ements satisfy | X j − X j − 1 | ≤ c j for suitable c j , and X j c 2 j ≤ (36 k 2 + 3 k (4 k + 1) 2 )∆ . 11 Pr o of. W e first consider a step in whic h w e color a v ertex w j ∈ V ( G ) \ T i . Let d j := | N ( w j ) ∩ T i | . Changing the color of w j can affect M i only if some neighbor of w j in T i receiv es one of the tw o colors in volv ed in the c hange. Since colors are uniform on [∆ − 1] , this o ccurs with conditional probabilit y at most 2 d j / (∆ − 1) ≤ 2 d j / ∆ . Moreo ver, by the additional requirement in the definition of M i that no color used on one triple app ears on an y other triple, recoloring w j / ∈ T i can change the indicator of at most one triple; hence it c hanges M i b y at most 1 . Therefore we may take c j ≤ 2 d j ∆ for w j / ∈ T i . Consequen tly , X w j / ∈ T i c 2 j ≤ 4 ∆ 2 X w j / ∈ T i d 2 j . Since | T i | = 3 k ∆ , w e hav e max j d j ≤ | T i | = 3 k ∆ and X w j / ∈ T i d j = e ( T i , V ( G ) \ T i ) ≤ ∆ | T i | = 3 k ∆ 2 . Hence X w j / ∈ T i d 2 j ≤ (max j d j ) X w j / ∈ T i d j ≤ (3 k ∆)(3 k ∆ 2 ) = 9 k 2 ∆ 3 , and th us X w j / ∈ T i c 2 j ≤ 4 ∆ 2 · 9 k 2 ∆ 3 = 36 k 2 ∆ . No w consider a step in which w e color a vertex w j ∈ T i . The v ariable M i dep ends explicitly on the colors inside T i (through the uniqueness constrain ts in its definition), so these steps are not negligible. Ho wev er, by construction, c hanging the color of a single v ertex of T i can affect the v alue of M i b y at most 2 , and hence we may take c j ≤ 2 for w j ∈ T i . Therefore X w j ∈ T i c 2 j ≤ X w j ∈ T i (2 × 2 k + 1) 2 = (4 k + 1) 2 | T i | = 3 k (4 k + 1) 2 ∆ . Com bining the tw o phases yields X j c 2 j = X w j / ∈ T i c 2 j + X w j ∈ T i c 2 j ≤ 36 k 2 ∆ + 3 k (4 k + 1) 2 ∆ = (36 k 2 + 3 k (4 k + 1) 2 )∆ , as required. Let B ( k ) := 36 k 2 + 3 k (4 k + 1) 2 and a ( k ) := (1 − ϵ ) k  3 5 − k  2 (1 − 3 k ) e − 5 , ϵ < 10 − 17 . Since µ ≥ a ( k )∆ , it suffices to solv e ( a ( k )∆ − 2) 2 2 B ( k )∆ ≥ 1 + ln  2(∆ 4 + 2)  . Optimising o ver k ∈ [10 − 7 , 1 / 9] , the minimum is attained at k ∗ ≈ 0 . 038895 , and the smallest in teger ∆ satisfying the ab ov e inequality is ∆ min = 7 , 327 , 700 , 972 . 12 4 A quadratic-time coloring algorithm In this section, we sho w Theorem 4.2 by giving a quadratic-time coloring algorithm for graphs with large maxim um degree. Our existence pro of uses the Lo v ász Lo cal Lemma to show that there is a random (∆ − 1) -coloring whose induced partial coloring (after deleting conflicts) a voids all bad ev ents in the family E := { A v : v ∈ L } ∪ { E i : S i is a near-clique } ∪ { F i : 1 ≤ i ≤ ℓ } . This step can b e made constructiv e using the resampling algorithm of Moser and T ardos [ 17 ]. Algorithmic Lo v ász Lo cal Lemma (Moser–T ardos). View the initial colors col( x ) ∈ [∆ − 1] as indep endent random v ariables, one p er vertex. Each bad even t B ∈ E dep ends only on the colors of a b ounded set of vertices, denoted vbl( B ) ⊆ V ( G ) . Starting from a uniformly random assignmen t col , whenev er some bad even t B o ccurs we resample all v ariables in vbl( B ) (i.e. redraw their colors indep enden tly and uniformly). The output assignment induces a partial coloring φ with no bad ev ents, and Algorithm 1 then extends φ deterministically to a prop er (∆ − 1) -coloring of G . Theorem 4.1 (Moser–T ardos [ 17 ]) . L et P b e a finite set of mutual ly indep endent r andom variables in a pr ob ability sp ac e, and let A b e a finite set of events determine d by these variables. Supp ose ther e exists an assignment of r e als x : A → (0 , 1) such that for every A ∈ A , P [ A ] ≤ x ( A ) Y B ∈ Γ( A ) (1 − x ( B )) , wher e Γ( A ) denotes the neighb orho o d of A in the dep endency gr aph. Then ther e exists an assignment of values to the variables in P that violates none of the events in A . Mor e over, the r andomize d r esampling algorithm r esamples e ach event A ∈ A at most an exp e cte d x ( A ) 1 − x ( A ) times b efor e it finds such an assignment. Conse quently, the exp e cte d total numb er of r esampling steps is at most X A ∈A x ( A ) 1 − x ( A ) . W e now describ e the algorithm that constructs the desired (∆ − 1) -coloring. Algorithm 1: (∆ − 1) -Coloring via Algorithmic Lo cal Lemma and Greedy Extension Input: A graph G with maximum degree ∆ , q ← ∆ − 1 , and the partition V ( G ) = L ∪ S ℓ i =1 S i where eac h S i is either a clique C i or a near-clique C i ∪ { v i } . Output: A prop er (∆ − 1) -coloring of G . 1: Define bad ev ents E ← { A v : v ∈ L } ∪ { E i : S i near-clique } ∪ { F i : 1 ≤ i ≤ ℓ } 2: F or each B ∈ E , define its v ariable set vbl( B ) 3: Initialize col : V ( G ) → [ q ] uniformly and indep endently 13 4: φ ← ConflictDelete ( col ) ▷ partial coloring after uncoloring conflicts 5: function ConflictDelete ( col ) 6: φ ← col 7: for eac h edge xy ∈ E ( G ) do 8: if col( x ) = col( y ) then 9: φ ( x ) ← ⊥ ; φ ( y ) ← ⊥ 10: end if 11: end for 12: return φ 13: end function 14: while some B ∈ E o ccurs under φ do 15: Cho ose an y such B 16: for eac h x ∈ vbl( B ) do 17: Resample col( x ) uniformly from [ q ] 18: end for 19: φ ← ConflictDelete ( col ) 20: end while ▷ A t this p oint, φ satisfies conditions (i)–(iii) b elo w. ▷ Greedy extension to a full (∆ − 1) -coloring (Lemma 3.1 ). 21: χ ← φ ▷ w e extend χ to all vertices 22: for eac h near-clique S i = C i ∪ { v i } do ▷ Condition (ii): v i has t wo uncolored neighbors in C i under χ 23: Let U i ← { x ∈ C i : χ ( x ) = ⊥} 24: Cho ose distinct u i , w i ∈ U i ▷ guaran teed by (ii) 25: χ ( v i ) ← an y color in [ q ] \ { χ ( x ) : x ∈ N ( v i ) , χ ( x )  = ⊥} 26: end for 27: for i = 1 to ℓ do ▷ Eac h S i is either C i or a near-clique; let C i b e its clique part 28: Let U i ← { x ∈ C i : χ ( x ) = ⊥} 29: Cho ose distinct w i , x i ∈ U i 30: for eac h y ∈ U i \ { w i , x i } do 31: χ ( y ) ← any color in [ q ] \ { χ ( z ) : z ∈ N ( y ) , χ ( z )  = ⊥} 32: end for 33: χ ( w i ) ← an y color in [ q ] \ { χ ( z ) : z ∈ N ( w i ) , χ ( z )  = ⊥} 34: χ ( x i ) ← an y color in [ q ] \ { χ ( z ) : z ∈ N ( x i ) , χ ( z )  = ⊥} 35: end for 36: for eac h vertex v ∈ L with χ ( v ) = ⊥ do 37: χ ( v ) ← any color in [ q ] \ { χ ( z ) : z ∈ N ( v ) , χ ( z )  = ⊥} 38: end for 39: return χ T ermination and exp ected n um b er of resamplings. W e use the standard witness form of the (symmetric) Lov ász Local Lemma. Assume P ( B ) ≤ p for all B ∈ E , and each B is dep endent with 14 at most d other ev ents in the dep endency graph. If ep ( d + 1) < 1 , then taking x := 1 / ( d + 1) gives p ≤ 1 e ( d + 1) ≤ x (1 − x ) d . By the main b ound in [ 17 ] (expected num b er of resamplings of each even t B is at most x/ (1 − x ) under suc h a witness assignment), the exp ected total num b er of resampling steps satisfies E [# resamplings ] ≤ X B ∈E x 1 − x ≤ |E | d . (6) (In particular, it is O ( |E | ) .) Exp ected running time. Let R := max B ∈E | vbl( B ) | . In our setting, each bad ev ent dep ends on a b ounded-radius neigh b ourho o d, hence R = p oly(∆) ; for example one ma y tak e R = O (∆ 2 ) for A v -t yp e and F i -t yp e ev ents (radius- 4 neighbourho o d), and similarly R = p oly (∆ 2 ) for E i . One resampling step recolors at most R v ertices and can b e chec k ed/up dated in p oly(∆) time (naiv ely O ( R ∆) using adjacency lists). Therefore the exp ected total running time is E [ T ] = O ( E [# resamplings ] · p oly (∆)) = O  |E | d · p oly(∆)  ⊆ p oly ( n, ∆) , (7) where n := | V ( G ) | and |E | = O ( n ) . Finally , the greedy extension phase in Algorithm 1 runs in O ( n ∆) time. By rep eatedly removing vertices of degree less than ∆ − 1 and cliques that do not satisfy the minimal coun ter example with section 2 , w e obtain a graph that meets the conditions of Algorithm 1 , which can then b e colored with ∆ − 1 colors. Finally , we reinsert the remo ved vertices in rev erse order of deletion and assign each a color according to the coloring scheme guaranteed by the lemmas in section 2 . Thus, the total exp ected time complexit y is O ( n 2 ) . Thus w e ha ve the follo wing theorem. Theorem 4.2. Ther e is a quadr atic-time (∆( G ) − 1) -c oloring algorithm for gr aphs with maximum de gr e e ∆( G ) at le ast 7 . 4 × 10 9 . Remark 4.3. While preparing this man uscript, w e learned ab out a recent pap er by Dvořák , Kang and Mikšaník [ 11 ], who established a more general coloring result in the setting of correspondence coloring. In particular, Theorem 4 in [ 11 ] can also imply the corresp onding existence statement for ordinary coloring under a weak er degree b ound. Ho wev er, our approac h is constructive. Our pro of yields an explicit quadratic-time algorithm for finding a (∆ − 1) -coloring for graphs with maxim um degree ∆ . A c kno wledgement This research was supp orted b y National Key R&D Program of China under Grant No. 2022YF A1006400 and National Natural Science F oundation of China under Grant No. 12571376. References [1] K. Azuma, W eigh ted sums of certain dep enden t random v ariables, T ohoku Math. J. 19 (1967), 357–367. 15 [2] O. V. Boro din and A. V. 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