Some Results on Cyclic Interval Edge Colorings of Graphs

TL;DR

This paper investigates cyclic interval edge colorings in bipartite and multipartite graphs, establishing new conditions under which such colorings exist, including for Eulerian bipartite graphs and certain biregular graphs, and confirms a conjecture for complete multipartite graphs.

Contribution

It provides new existence results for cyclic interval colorings in bipartite, Eulerian, biregular, and complete multipartite graphs, including confirming a conjecture for the latter.

Findings

Bipartite graphs with even maximum degree ≥4 admit cyclic interval Δ(G)-colorings under certain degree conditions.

Eulerian bipartite graphs with maximum degree ≤8 have cyclic interval colorings.

All complete multipartite graphs have cyclic interval colorings.

Abstract

A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{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 $G$ with even maximum degree $\Delta(G)\geq 4$ admits a cyclic interval $\Delta(G)$-coloring if for every vertex $v$ the degree $d_G(v)$ satisfies either $d_G(v)\geq \Delta(G)-2$ or $d_G(v)\leq 2$. We also prove that every Eulerian bipartite graph $G$ with maximum degree at most $8$ 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 ($r\geq 2$) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this settles in the affirmative, a conjecture of Petrosyan and Mkhitaryan.