Interval edge-colorings of cubic graphs

Interval edge-colorings of cubic graphs Petros A. Petrosyan Institute for Informatics and Automation Problems of NAS of RA, Department of Informatics and Applied Mathematics, Y SU, Yerevan, Arm enia e-mail: pet _petros@ipi a.sci.am ABSTRACT An edge-coloring of a multigrap h with colors 1, is called an interval coloring if all colors are used, and the colors of edges incid ent to any vertex of G are distinct and form an interval of integers. In this paper we prove that if is a connected cubic multigraph (a connected cubic graph) that admits an interval coloring, then G 2 , , t … t − G t − () 1 VG t + ≤ ( () VG t ≤ ), where is the s et of vertices of Moreover, if G is a connected cubic graph , , and has an interval t coloring , then () VG . G 4 GK ≠ G − () 1 . VG t − ≤ We also show that these upper bounds are sh arp. Finally, we prove that if is a bipartite subcubic multigraph, then G has an interval edge-coloring with no more than four colors. G Keywords Edge-coloring, interval edge- c oloring, cubic graph, cubic multigraph, bipartite graph. 1. INTRODUCTION In this paper we consider graphs which are finite, undirected, and have no loops or multiple edges and multigraphs which may contain multiple edg es but no loops. Let and denote the sets of vertices and edges of a graph , respecti vely. An biregular bipartite graph is a bipartite graph with the vertices in one part all having degree and the vertices in the other part all having degree . The degree of a vertex is denoted by , the maximum degree of a vertex in G by () VG () EG G (, ) ab − G G a b () vV G ∈ ( ) G dv () G Δ and the chromatic index of G by () G χ ′ . The terms and concepts that we do not define can be found in [7,10]. An interval t − coloring of a multigraph is an edge- coloring of with colors 1, such that at least one edge of is colored by color , and the edges incident to e ach vertex are colored by consecutive colors. A m ultigraph G is interval- colorable if there is for which has an interval coloring. The set of all interval-colorab le multigraphs is denoted by For a multig raph , the least and the greatest values of for which has an interval G G 2 , , t … G , 1 ,2 , , ii t = … () vV G ∈ ( ) G dv 1 t ≥ G t − . N G ∈ N t G t − coloring are denoted by and , respectively . () wG () WG The concept of interval edg e-coloring of multigraphs was introduced by Asratian and Kamalian [4]. In [4,5] , they proved the following two th eorems. Theorem 1 . If G is a regular graph , then G ∈ N if and only if () ( ) GG χ ′ = Δ . Theorem 2. If G is a connected triangle-free graph and G ∈ N , then () 1 . () WG VG − ≤ Corollary 1. If is a connected bipartite graph and G G ∈ N , then () 1 . () VG WG − ≤ Note that this upper bound is tight for complete bipartite graphs , mn K , since and , mn K ∈ N () , 1 mn K Wm n = +− [11]. Nevertheless, for some bipartite graphs this upper bound can be improved. Recently, Asratian and Casselgren [3] proved the following Theorem 3. If G is a connected ( biregular bipartite graph with , ) ab − ( ) () 2 VG a b + ≥ and , then G ∈ N () 3 . () VG WG − ≤ For general graphs, Kamalian proved th e following Theorem 4. [12] If is a conn ected graph and G G ∈ N , then () 3 . () 2 VG WG − ≤ Note that the upper bound is tight for 2 K , but if , then this upper bound can be improved. 2 GK ≠ Theorem 5. [8] If is a connected graph with G () 3 VG ≥ and G ∈ N , then () 4 . () 2 VG WG − ≤ For regular grap hs, Kamalian and Petrosyan proved the following Theorem 6. [13] If is a connected regular graph with G r − () 22 VG r ≥+ and G ∈ N , then () 5 . () 2 VG WG − ≤ On the other hand, in [16], Petrosyan proved the following theorem. Theorem 7. For any 0 ε > , there is a connected graph such that G G ∈ N and ( ) 2( () VG WG ε − ≥ ) . For planar graphs, the coefficient in upper bounds of Theorems 4-6 was improved by Axenovich. In [6], she proved the following Theorem 8. If is a connected planar graph and G G ∈ N , then 11 () . 6 () VG WG ≤ In this paper we investigate interval edge-color ings of cubic graphs and multigraphs. W e also consider interval edg e- colorings of bipartite subcubi c multigraphs . 2. MAIN RESULTS First, we give an upper bound on for interval- colorable connected cubic multigraphs () WG . G Theorem 9. If is a connected cubic multigraph and , then G G ∈ N () 1 . () VG WG + ≤ Note that the upper bound in Theorem 9 is tight. The following theorem holds. Theorem 10. For a ny there exists a connected cubic multigraph G with 2, n ≥ () 2 VG n = such that G ∈ N and () 1 . () VG WG + = Next, we show that if is a connected cubic graph and , then G G ∈ N () . () VG WG ≤ Theorem 11. If is a connected cubic graph and G G ∈ N , then () . () VG WG ≤ Moreover, if , then 4 GK ≠ () 1 . () VG WG − ≤ Note that the upper bound in Theorem 11 is tight, too. The following theorem holds. Theorem 12. For a ny there exists a connected cubic graph G with 3, n ≥ () 2 VG n = such that G and ∈ N () 1 . () VG WG − = It is well-known that th e four color theorem [1,2] is equivalent to the statement that e very bridgeless cubic planar graph is 3 -edge-colorable [10]. From here and taking into account Theorem 1, we obtain the following Theorem 13. If is a bridgeless planar cubic graph, then and G G ∈ N () 3 . wG = 4 K is an example of bridgeless planar cubic graph with 4 4 () 4 , () VK WK = = but if G is a 2-connected plan ar cubic graph and , then, by Th eorem 11, we have 4 GK ≠ () 1 . () VG WG − ≤ Moreover, we show that the following theorem holds. Theorem 14. For any there exists a 2 -connected planar cubic graph with 3, n ≥ G () 2 VG n = such that () 1 . () VG WG − = We also consider cubic Hali n graphs. A Halin graph is a planar graph constructed from a plane embedding of a tree with at least fou r vertices and wi th no vertices of degree 2, b y connecting all the leaves of the tree with a cycle , traversing the leaves in the order given by planar embedding of the tree . In particular, we proved the following result. Theorem 15. If is a cubic Halin graph, then G G ∈ N and () 3 . wG = Moreover, for any there exists a cubic Halin graph G with 2, n ≥ () 2 VG n = such that () 2. 2 () VG WG + ≥ However, there are interv al-colorable cubic graphs for which the value of the parameter is close to G () WG () . 2 VG For example, in [15], it was proved that for Möbius ladders 2 n M ( ) with vertices the following theorem holds. 2 n ≥ 2 n Theorem 16. For any 2, n ≥ 1. 2 , n M ∈ N 2. ( ) 2 3, n wM = 3. ( ) 2 2, n WM n = + 4. if ( )( 2 , n wM t W M ≤≤ ) 2 n then the cubic graph 2 n M has an interval t coloring. − In [14], a similar result was proved for n − prism graphs ( ) with vertices. In particular, Khchoyan [14] proved the following 2 n CK  3 n ≥ 2 n Theorem 17. For any 2, n ≥ 1. 2 , n CK  ∈ N 2. ( ) 2 3, n wC K  = 3. ( ) 2 2, n WC K n  = + 4. if ( )( 2 , n wC K t W C K  ≤≤ ) 2 n  then the cubic graph has an interval coloring. 2 n CK  t − In the same work [14], the author considered a ring of diamonds (a ring with k diamonds and ). The graph k D 2 k ≥ 4 e K − is called a diamond, and a ring of diamonds is a sequence of diamonds in which consecutive diamonds are connected. Clearly, a ring of diamonds is a 2- k k k D connected planar cubic graph. For these graphs, Khchoyan proved the follow ing Theorem 18. For any 2, k ≥ 1. , k D ∈ N 2. () 3, k wD = 3. () 3 4, 2 k k WD + ≥ if is even, k () 32 2 k k WD ⎡⎤ , + ⎢⎥ ⎢⎥ ≥ k if is odd, 4. if is even and k () 3 4, 2 k k wD t + ≤≤ then has an interval t k D − coloring, if is odd and k () 3 2 k k wD t ⎡⎤ 2 , + ⎢⎥ ⎢⎥ ≤≤ then has an interval t k D − coloring. Finally, we consider bipartit e multigraphs. In [9], Hansen proved the following theor em for bipartite subcubic gr aphs. Theorem 19. If is a bipartite graph with G () 3 G Δ ≤ , then and G ∈ N () 4 . wG ≤ We show that this theorem also holds for bipartite subcubic multigraphs. More precisely, the following theorem holds. Theorem 20. If is a bipartite multigraph with G () 3 G Δ ≤ , then G and ∈ N () 4 . wG ≤ By Theorems 9 and 20, we obtain the following Corollary 2. If is a connected bipart ite cubic multigraph, then and G G ∈ N () 1 . () VG WG + ≤ Moreover, for an y there exists a connected bipart ite cubic multigraph with 2, n ≥ G () 2 VG n = and () 1 . () VG WG + = On the other hand, by Theorems 3 and 19, we obtain the following Corollary 3. If is a connected bipartite cubic g raph with G () 12 VG ≥ , then and G ∈ N () 3 . () VG WG − ≤ ACKNOWLEDGEMENT We would like to express our gratitude to H . Tananyan and V. Mkrtchyan for useful discussions on the subject. REFERENCES [1] K. Appel, W. 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