Examples of cyclically-interval non-colorable bipartite graphs

Abstract. For an undirected, simple, finite, connected graph G, we denote by V(G)and E(G) the sets of its vertices and edges, respectively. A function ϕ : E(G) →{1,2,...,t} is called a proper edge t-coloring of a graph G if adjacent edges arecolored differently and each of t colors is used. An arbitrary nonempty subset ofconsecutive integers is called an interval. If ϕ is a proper edge t-coloring of a graphG and x ∈ V(G), then S G (x,ϕ) denotes the set of colors of edges of G which areincident with x. A proper edge t-coloring ϕof a graph Gis called a cyclically-intervalt-coloring if for any x∈ V(G) at least one of the following two conditions holds: a)S G (x,ϕ) is an interval, b) {1,2,...,t}\S G (x,ϕ) is an interval. For any t∈ N, let M t be the set of graphs for which there exists a cyclically-interval t-coloring, and letM≡[ t≥1 M t .Examples of bipartite graphs that do not belong to the class Mare constructed.Mathematics subject classification: 05C15.Keywords and phrases: cyclically-interval edge coloring, bipartite graph.