Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph G with colors 1,2,⋯,t is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree Δ(G)≥4 admits a cyclic interval Δ(G)‐coloring if for every vertex v the degree dG(v) satisfies either dG(v)≥Δ(G)−2 or dG(v)≤2. We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for (a,b)‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all (2r−2,2r)‐biregular (r≥2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.