Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

TL;DR

This paper investigates bounds on the minimum and maximum number of colors needed for interval and cyclic interval edge colorings of multigraphs, providing new sharp bounds and partial results towards a broader conjecture.

Contribution

It introduces new sharp bounds on the color counts for multigraphs with interval and cyclic interval colorings, including specific bounds for 2-connected multigraphs and partial results for the conjecture on triangle-free graphs.

Findings

For 2-connected multigraphs with an interval coloring, W(G) is bounded by 1 + floor(|V(G)|/2)*(Δ(G)-1).

The paper provides partial evidence supporting the conjecture that W_c(G) ≤ |V(G)| for triangle-free graphs with cyclic interval colorings.

The conjecture holds for graphs with maximum degree at most 4.

Abstract

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors, under the condition that color $1$ is considered as consecutive to color $t$. Denote by $w(G)$ ($w_{c}(G)$) and $W(G)$ ($W_{c}(G)$) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph $G$, respectively. We present some new sharp bounds on $w(G)$ and $W(G)$ for multigraphs $G$ satisfying various conditions. In particular, we show that if $G$ is a $2$-connected multigraph with an interval coloring, then $W(G)\leq 1+\left\lfloor \frac{|V(G)|}{2}\right\rfloor(\Delta(G)-1)$. We also give several results towards the general conjecture that $W_{c}(G)\leq |V(G)|$ for any triangle-free graph $G$ with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most $4$.