On coloring of graphs of girth 2l + 1 without longer odd holes

Abstract

A hole is an induced cycle of length at least 4. Let l ≥ 2 be a positive integer, let G l denote the family of graphs which have girth 2l + 1 and have no holes of odd length at least 2l + 3, and let G ∈ G l . For a vertex u ∈ V (G) and a nonempty set] is bipartite for each i > 0, and consequently χ(G) ≤ 4, where G[S] denotes the subgraph induced by S. Let θ − be the graph obtained from the Petersen graph by deleting three vertices which induce a path, let θ + be the graph obtained from the Petersen graph by deleting two adjacent vertices, and let θ be the graph obtained from θ + by removing an edge incident with two vertices of degree 3. For a graph G ∈ G 2 , we show that if G is 3-connected and has no unstable 3-cutset then G must induce either θ or θ − but does not induce θ + . As corollaries, χ(G) ≤ 3 for every graph G of G 2 that induces neither θ nor θ − , and minimal non-3-colorable graphs of G 2 induce no θ + .

Graphs with girth 9 and without longer odd holes are 3‐colourable

“…Xu, Yu and Zha [16] proved that every pentagraph is 4-colourable. Generalising the result of [16], Wu, Xu and Xu [14] proved that graphs in l l 2  ≥  are 4-colourable and proposed the following conjecture.…”