Candy-passing Games on General Graphs, II

CAND Y-P ASSING GAMES ON GENERAL GRAPHS, I I P AUL MYER KOMINERS AND SCOTT DUKE KOMINERS Abstract. W e give a new proof that an y candy-passing game on a graph G with at least 4 | E ( G ) | − | V ( G ) | candies stabilizes. Unlike the prior l iterature on candy-passing games, we use metho ds f rom the general theory of chip-firing games whic h al l o w us to obtain a p olynomial b ound on the num ber of rounds befor e stabilization. 1. Introduction W e let G b e an undirected graph and resp ectiv ely de no te the vertex and edge sets of G by V ( G ) and E ( G ). The c andy-p assing game on G is defined by the following rules : • At the beg inning o f the game, c > 0 candies a re distributed among | V ( G ) | student s, ea c h of whom is seated a t some distinct v ertex v ∈ V ( G ). • A whistle is sounded at a r e g ular interv al. • Each time the whistle is sounded, every studen t who is a ble to do so pa s ses one c andy to each o f his neig h b ors. (If a t the beginning of th is step a student ho lds fewer ca ndies than he has neighbors, he do es nothing.) T anton [6] intro duced this game for cy clic G . The author s [4] extended the game to general graphs G . The ca ndy-passing game o n G is a sp ecial cas e of the w ell-known chip-firing game on G introduced b y Bj¨ orner, Lov´ asz, and Shor [2]. F urthermo re, terminating candy-passing g a mes on G are actually equiv alent to terminating chip-firing g ames on G , by the following key theorem: Theorem 1 ([2]) . The initial c onfigur ation of a chip-firing game on G determines whether the game wil l terminate. If the game do es terminate, then b oth the final c onfigu r ation and length of the game ar e dep endent only on the initial c onfigur ation. T ermina ting c hip-firing games hav e been s tudied extensively and are surprisingly well-behav ed. In a ddition to Theorem 1, it is kno wn that terminating chip-firing pro cesesses finish in po lynomial time (see [7]). Chip-firing games also have imp or- tant applications; notably , they are re la ted to T utte p olynomials (see [5]) and the critical gro ups of gr aphs (see [1]). Infinite chip-firing ga mes have received less attention, as the notio n o f a n “ end state” of suc h a ga me is ambiguous. By contrast, an infinite candy-passing game admits a clear s tabilization condition: the game is said to hav e stabilize d if the configuratio n of candy will ne ver again change. 2000 Mathematics Subje ct Classific ation. 05C35, 05C8 5, 68Q25 (Primary); 37B15, 68R10, 68Q80 (Secondary). Key wor ds and phr ases. candy-passing, c hip-firing, gr aph game, stabilization, p olynomial time. The second autho r gratefully ackno wl edges the support of a Har v ard Mathe matics Departmen t Highbridge F ell o wship. 1 2 P AUL MYER KOMINERS AND SCOTT DUKE KOMINERS The fir s t author [3] studied the end behavior of candy-passing games on n - cycles, proving the eventual stabilization of any candy-passing game on an n -cycle with at least 3 n − 2 candies. The a uthors [4] extended this analysis to arbitra r y connected gra phs G , showing that any candy-pass ing game on suc h G with a t least 4 | E ( G ) | − | V ( G ) | candies will stabilize. Here, we giv e a new pro of o f the s tabilization result for g eneral connected graphs, using methods which allow us to obtain a p olynomial bound on the stabilization time. Our appro ach draws from the literature on chip-firing, using in pa rticular a key result from T ardo s’s [7] pro of that terminating chip-firing games conclude in po lynomial time. 2. The Setting As in the earlier work on candy-passing games, w e refer to the interv al b et ween soundings of the whistle as a r ound of candy - passing. W e deno te by ϕ t ( v ) the tota l nu mber times a vertex v ∈ V ( G ) ha s passed candy by the e nd of round t . Since infinite candy- pa ssing g a mes differ from infinite chip-firing g ames, w e will contin ue to distinguish b et ween “candies” and “chips.” How ev er, w e drop the student metaphor, treating the candy piles as b elonging to the vertices of the graph G . F or consistency , we denote the tota l num ber of ca ndies in a candy- passing ga me by c throughout. Abusing terminolo g y slight ly , w e say that a vertex has stabilize d in some round if, after that round, the a moun t of candy held by that vertex will not change during the remainder of the g ame. F or a vertex v ∈ V ( G ), we denote the degree o f v by deg( v ). W e say that a vertex v ∈ V ( G ) is abundant if it ho lds at least 2 deg( v ) pieces of c andy . An y vertex v ∈ V ( G ) with k ≥ deg ( v ) ca ndies at the b eginning of a r ound pas ses deg( v ) pieces o f ca ndy to its neighbors and ca n, at most, receive one piece o f ca ndy from each of its deg ( v ) neighbo rs. Thus, such a vertex cannot end the round with more than k candies. In particular , then, the set o f abundant vertices of G can only shrink ov er the course of a ca ndy-passing game on G . 3. Main Theorem W e will prov e the follo wing stabilization theorem: Theorem 2. L et G b e a c onne ct e d gr aph with diameter d . In any c andy-p assing game on G with c ≥ 4 | E ( G ) | − | V ( G ) | c andies, every vertex v ∈ V ( G ) wil l stabili ze within | V ( G ) | · d · c r ounds. The stabilization comp onent of Theor em 2 w as o btained in [4, Theo rem 2]. Our metho ds a re inspir ed b y those o f T ardo s [7]; they are essentially independent of the arguments used in [3 ] and [4]. W e use the following lemma, which is a specia l case of T a rdos’s [7 ] Lemma 5: Lemma 3. L et v , v ′ ∈ V ( G ) b e adjac ent vertic es of G . T hen, | ϕ t ( v ) − ϕ t ( v ′ ) | ≤ c for al l t . Additionally , we need an obser v ation ab out the condition c ≥ 4 | E ( G ) | − | V ( G ) | . CAND Y-P ASS ING GAMES ON GENERAL GRAPHS, I I 3 Lemma 4. F or G a gr aph and c ≥ 4 | E ( G ) | − | V ( G ) | , in any chi p-firing game on G with c c andies ther e is at le ast one ve rtex v ∗ ∈ V ( G ) which p asses c andy ev ery r oun d. Pr o of. It s uffices to find a vertex v ∗ ∈ V ( G ) whic h passes candy every round t during which so me v ertex v ∈ V ( G ) holds few er than 2 deg( v ) − 1 candies. As observed ab ov e, it is not p ossible for a vertex v ∈ V ( G ) whic h is not abundant at the b eginning of ro und t t o b ecome a bundan t after round t . How ev er, the condition c ≥ 4 | E ( G ) | − | V ( G ) | guarantees that whenever some v ∈ V ( G ) holds few er than 2 deg( v ) − 1 candies there is also at le a st o ne abundant vertex v ′ ∈ V ( G ) . The existence of s o me vertex v ∗ ∈ V ( G ) which is a bundan t in every round when some vertex v ∈ V ( G ) has fewer than 2 deg( v ) − 1 candies then follows immediately .  Remark. Lemma 4 is, in some sense, dual to T a rdos’s [7] Le mma 4 which s ho ws that for any terminating chip-firing game on G there is a distinguis hed vertex v ∗ ∈ V ( G ) whic h never fires. W e may now pro ceed with the pro of of our main result: Pr o of of The or em 2. By Lemma 4, there is some v ertex v ∗ ∈ V ( G ) whic h passes candy every round. Denoting the rounds by t = 1 , 2 , . . . , we then hav e ϕ t ( v ∗ ) = t for all rounds t . By Lemma 3 , we then know that | ϕ t ( v ∗ ) − ϕ t ( v ) | ≤ d · c for all t and v ∈ V ( G ). Since ϕ t ( v ∗ ) is strictly increasing in t , no v ∈ V ( G ) may fail to pas s candy for more than d · c r ounds. In the w orst case, all but one vertex pass candy in each round when s ome vertex do es not pass ca ndy ; hence after | V ( G ) | · d · c rounds all the vertices of G pass candy every round.  References [1] N. L. Biggs, Chip-firi ng and the critical group of a graph, Journal of Algebr aic Combinatorics 9 (1), 1999, pp. 25–45. [2] A. Bj¨ orner, L. Lo v´ asz, and P . Shor, Chip-firing games on graphs, Eur op e an Journal of Com- binatorics 1 2 (4), 1991, pp. 283–291. [3] P . M. Kominers, The candy-passing game for c ≥ 3 n − 2, Pi Mu Epsilon Journal 12 (8), 2008, pp. 459–460. [4] P . M. Kominers and S. D. Kominers, Candy-passing games on general graphs, I, arXiv:08 07.4450 . [5] C. M. L´ op ez, Chi p firing and the T utte p olynomial, Anna ls of Combinatorics 1 (1), 1997, pp. 253–259. [6] J. T anton, T o da y’s puzzler, The St. Mark’s Institute of Mathematics Newsletter , N ov ember 2006. [7] G. T ardos, Polynomial b ound for a c hip firing game on graphs, SIAM Journal on D iscr ete Mathematics 1 (3), 1998, pp. 397–398. Student, Depar tment of Ma themat ics, Massachusetts In stitute of Technology E-mail addr ess : pkoms@mit.e du Student, Depar tment of Ma themat ics, Har v ard University c/o 8520 Burnin g Tree Road Bethesda, MD 2 0817 E-mail addr ess : kominers@fa s.harvard.edu