TY - JOUR

T1 - Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

AU - Kierstead, Henry

AU - Kostochka, Alexandr V.

N1 - Funding Information:
* Research of this author is supported in part by the NSA grant MDA 904-03-1-0007. † Research of this author is supported in part by the NSF grant DMS-0650784 and by grant 06-01-00694 of the Russian Foundation for Basic Research.

PY - 2010

Y1 - 2010

N2 - Chen, Lih, and Wu conjectured 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. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks' Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains Kr,r and V(G) partitions into subsets V0,..., Vt such that G[V0] = Kr,r and for each 1 ≤ i ≤ t, G[Vi] = Kr. We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.

AB - Chen, Lih, and Wu conjectured 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. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks' Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains Kr,r and V(G) partitions into subsets V0,..., Vt such that G[V0] = Kr,r and for each 1 ≤ i ≤ t, G[Vi] = Kr. We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.

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U2 - 10.1007/s00493-010-2420-7

DO - 10.1007/s00493-010-2420-7

M3 - Article

AN - SCOPUS:77956858157

VL - 30

SP - 201

EP - 216

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -