A refinement of a result of Corrádi and Hajnal

Corrádi and Hajnal proved that for every k ≥ 1 and n ≥ 3k, every n-vertex graph with minimum degree at least 2k contains k vertex-disjoint cycles. This implies that every 3k-vertex graph with maximum degree at most k − 1 has an equitable k-coloring. We prove that for s∈{3,4} if an sk-vertex graph G with maximum degree at most k has no equitable k-coloring, then G either contains Kk+1 or k is odd and G contains Kk,k. This refines the above corollary of the Corrádi-Hajnal Theorem and also is a step toward the conjecture by Chen, Lih, and Wu that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1.

[1]  Ko-Wei Lih,et al.  On equitable coloring of bipartite graphs , 1996, Discret. Math..

[2]  Noga Alon,et al.  Spanning subgraphs of random graphs , 1992, Graphs Comb..

[3]  Noga Alon,et al.  AlmostH-factors in dense graphs , 1992, Graphs Comb..

[4]  A. V. Kostochkaa,et al.  On equitable -coloring of graphs with low average degree , 2005 .

[5]  Alexandr V. Kostochka,et al.  A list analogue of equitable coloring , 2003 .

[6]  Zhihui Xue,et al.  Review of Scheduling computer and manufacturing processes by Jacek Blazewicz, Klaus H. Ecker, Erwin Pesch, Guenter Schmidt, Jan Weglarz Springer-Verlag 2001 , 2003 .

[7]  P. Erdos,et al.  On the maximal number of independent circuits in a graph , 1963 .

[8]  Vojtech Rödl,et al.  Perfect Matchings in ε-Regular Graphs and the Blow-Up Lemma , 1999, Comb..

[9]  Alexandr V. Kostochka,et al.  A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring , 2008, Combinatorics, Probability and Computing.

[10]  Sriram V. Pemmaraju,et al.  Equitable colorings extend Chernoff-Hoeffding bounds , 2001, SODA '01.

[11]  Ko-Wei Lih,et al.  Equitable Coloring and the Maximum Degree , 1994, Eur. J. Comb..

[12]  P. Lax.  Proof of a conjecture of P. Erdös on the derivative of a polynomial , 1944 .

[13]  Alexandr V. Kostochka,et al.  Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture , 2010, Comb..

[14]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[15]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[16]  Vivek Sarin,et al.  Domain Decomposition , 2011, Encyclopedia of Parallel Computing.

[17]  Alexandr V. Kostochka,et al.  On equitable Delta-coloring of graphs with low average degree , 2005, Theor. Comput. Sci..

[18]  Alexandr V. Kostochka,et al.  Surveys in Combinatorics 2009: Extremal graph packing problems: Ore-type versus Dirac-type , 2009 .

[19]  Alexandr V. Kostochka,et al.  Every 4-Colorable Graph With Maximum Degree 4 Has an Equitable 4-Coloring , 2012, J. Graph Theory.

[20]  Colva M. Roney-Dougal,et al.  Surveys in combinatorics 2009 , 2009 .

[21]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[22]  Bolyai János Matematikai Társulat,et al.  Combinatorial theory and its applications , 1970 .

[23]  Endre Szemerédi,et al.  A fast algorithm for equitable coloring , 2010, Comb..

[24]  Alexandr V. Kostochka,et al.  An Ore-type theorem on equitable coloring , 2008, J. Comb. Theory, Ser. B.

[25]  Svante Janson,et al.  The infamous upper tail , 2002, Random Struct. Algorithms.

[26]  Klaus H. Ecker,et al.  Scheduling Computer and Manufacturing Processes , 2001 .