Odd and even cycle lengths, minimum degree and chromatic number in graphs
Abstract
In this paper, we prove similar results for odd and even cycle lengths. Let Lo(G) denote the set of odd cycle lengths of G and o(G) denote the longest odd cycle length. In 1992, Gy\'arf\'as proved that (G)≤ 2|Lo(G)|+2, and if w(G)≤ 2|Lo(G)|+1, then (G)≤ 2|Lo(G)|+1. We first prove that if G is a 2-connected non-bipartite graph with δ(G)≥ 2k, then |Lo(G)|≥ k. Moreover, if |Lo(G)|=k, then 2|Lo(G)|+1=o(G), and either K2k+1⊂eq G or (G)≤ 2k. Applying this result, we prove that if w(G)≤ 2|Lo(G)|, then (G)≤ 2|Lo(G)| for |Lo(G)|≥ 2, improving the result of Gy\'arf\'as. We also construct a class of graphs with w(G)=2|Lo(G)|-1 but (G)=2|Lo(G)| for every |Lo(G)|≥ 2. Using our result, we give a short proof of a similar result of (G) and o(G) proved by Kenkre and Vishwanathan. Our second part is about even cycle lengths. Let Le(G) denote the set of even cycle lengths of G and e(G) denote the longest even cycle length. In 2004, Mih\'ok and Schiermeyer proved that (G)≤ 2|Le(G)|+3, and if w(G)≤ 2|Le(G)|+2, then (G)≤ 2|Le(G)|+2. We first prove that if G is a 2-connected graph with δ(G)≥ 2k+1, then |Le(G)|≥ k. Moreover, if |Le(G)|=k, then 2|Le(G)|+2=e(G), and either K2k+2⊂eq G or (G)≤ 2k+1. Applying this result, we prove that if w(G)≤ 2|Le(G)|+1, then (G)≤ 2|Le(G)|+1 for |Le(G)|≥ 2, improving the result of Mih\'ok and Schiermeyer. We also construct a class of graphs with w(G)=2|Le(G)| but (G)=2|Le(G)|+1 for every |Le(G)|≥ 2. Our result can deduce a similar result of (G) and e(G). The above results also improve some results of consecutive odd or even cycle lengths.
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