Covering line graphs with equivalence relations

TL;DR

This paper investigates the equivalence number of line graphs, providing new bounds that disprove a recent conjecture and exploring related invariants, with implications for computational complexity.

Contribution

It establishes improved bounds for the equivalence number of line graphs and shows NP-completeness of deciding certain related invariants even in triangle-free graphs.

Findings

Bounds for $ ext{eq}(L(G))$ in terms of $ ext{chi}(G)$

Disproof of a conjecture about triangle-free graphs

NP-completeness of deciding $ ext{sigma}(G) \

Abstract

An equivalence graph is a disjoint union of cliques, and the equivalence number $\mathit{eq}(G)$ of a graph $G$ is the minimum number of equivalence subgraphs needed to cover the edges of $G$. We consider the equivalence number of a line graph, giving improved upper and lower bounds: $\frac 13 \log_2\log_2 \chi(G) < \mathit{eq}(L(G)) \leq 2\log_2\log_2 \chi(G) + 2$. This disproves a recent conjecture that $\mathit{eq}(L(G))$ is at most three for triangle-free $G$; indeed it can be arbitrarily large. To bound $\mathit{eq}(L(G))$ we bound the closely-related invariant $\sigma(G)$, which is the minimum number of orientations of $G$ such that for any two edges $e,f$ incident to some vertex $v$, both $e$ and $f$ are oriented out of $v$ in some orientation. When $G$ is triangle-free, $\sigma(G)=\mathit{eq}(L(G))$. We prove that even when $G$ is triangle-free, it is NP-complete to decide whether or not $\sigma(G)\leq 3$.