On odd colorings of sparse graphs

Abstract

An oddc-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times within its open neighborhood. A proper conflict-freec-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing exactly once within its neighborhood. Clearly, every proper conflict-free c-coloring is also an odd c-coloring. Cranston conjectured that every graph G with maximum average degree mad(G)<4cc+2 (where c≥4) has an odd c-coloring, and he proved this conjecture for c∈{5,6}. Note that the bound 4cc+2 is best possible. Cho et al. solved Cranston’s conjecture for c≥5, strengthening the result by transitioning from odd c-coloring to proper conflict-free c-coloring. However, they did not provide all the extremal non-colorable graphs G with mad(G)=4cc+2, which remains an open question of interest.In this paper, we tackle this intriguing extremal problem. We aim to characterize all non-proper conflict-free c-colorable graphs G with mad(G)=4cc+2. For the case of c=4, Cranston’s conjecture is not true, as evidenced by the existence of a counterexample: a graph whose every block is a 5-cycle. Cho et al. proved that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. We improve this result by proving that a graph G with mad(G)≤229 (with equality allowed) is not odd 4-colorable if and only if G belongs to a specific class of graphs. On the other hand, Cho et al. established that a planar graph with girth at least 5 has an odd 6-coloring; we improve it by proving that a planar graph without 4−-cycles adjacent to 7−-cycles also has an odd 6-coloring.