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 degreed G ( v )satisfies eitherd G ( v ) ≥ Δ ( G ) − 2 ord G ( 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 ( 2 r − 2 , 2 r ) ‐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.