On cyclically-interval edge colorings of trees

Abstract. For an undirected, simple, finite, connected graph G, we denote by V ( )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 ϕ of a graph G is 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 .For an arbitrary tree G, it is proved that G ∈ Mand all possible values of t are foundfor which G ∈ M t .Mathematics subject classification: 05C05, 05C15.Keywords and phrases: tree, interval edge coloring, cyclically-interval edge color-ing.