#### Title:

Interval cyclic edge-colorings of graphs

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#### Uncontrolled Keywords:

Edge-coloring ; interval coloring ; interval cyclic coloring ; bipartite graph ; complete graph

#### Abstract:

A proper edge-coloring of a graph G with colors 1, . . . , t is called an interval cyclic t-coloring if all colors are used, and the edges incident to each vertex v ∈ V (G) are colored by dG(v) consecutive colors modulo t, where dG(v) is the degree of a vertex v in G. A graph G is interval cyclically colorable if it has an interval cyclic t-coloring for some positive integer t. The set of all interval cyclically colorable graphs is denoted by Nc. For a graph G ∈ Nc, the least and the greatest values of t for which it has an interval cyclic tcoloring are denoted by wc(G) and Wc(G), respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if G is a triangle-free graph with at least two vertices and G ∈ Nc, then Wc(G) ≤ |V (G)| + ∆(G) − 2. We also obtain bounds on wc(G) and Wc(G) for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.

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#### Affiliation:

Yerevan State University ; Department of Informatics and Applied Mathematics ; Institute for Informatics and Automation Problems

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#### Conference title:

7th Cracow Conference on Graph Theory Rytro