Every toroidal graph without 3-cycles is odd 7-colorable

2022/06/13 by Tian, Fangyu, Yin, Yuxue
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.06052

Abstract

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors k that can ensure an odd coloring of a graph G is denoted by χ_o(G). We say G is k-colorable if χ_o(G)≤ k. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if G is planar then χ_o(G) ≤ 9 . A toroidal graph is a graph that can be embedded on a torus. Note that a K₇ is a toroidal graph, χ_o(G)≤7. In this paper, we proved that, every toroidal graph without 3-cycles is odd 7-colorable. Thus, every planar graph without 3-cycles is odd 7-colorable holds as a corollary. That's to say, every toroidal graph is 7-colorable can be proved if the remained cases around 3-cycle is resolved.

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