A note on one-sided interval edge colorings of bipartite graphs

TL;DR

This paper proves a conjecture that the minimum number of colors needed for an $X$-interval edge coloring in bipartite graphs with bounded degree can be bounded by a cubic polynomial, improving understanding of such colorings.

Contribution

The paper confirms the conjecture that a cubic polynomial bounds the $X$-interval chromatic number for bipartite graphs with bounded maximum degree, providing new upper bounds.

Findings

Proved the conjecture that a cubic polynomial bounds $ ext{chi'}_{int}(G,X)$.

Established improved upper bounds for graphs with small maximum degree.

Demonstrated that a cubic polynomial suffices for the bound.

Abstract

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that there is a polynomial $P(x)$ such that if $G$ has maximum degree at most $\Delta$, then $\chi'_{int}(G,X) \leq P(\Delta)$. In this short note, we prove this conjecture; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on $\chi'_{int}(G,X)$ for bipartite graphs with small maximum degree.