Covering line graphs with equivalence relations

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 $\sigma(G)\leq 3$.Comment: 10 pages, submitted in July 200

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure