Track number of line graphs

TL;DR

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Contribution

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Abstract

The track number $\tau(G)$ of a graph $G$ is the minimum number of interval graphs whose union is $G$. We show that the track number of the line graph $L(G)$ of a triangle-free graph $G$ is at least $\lg \lg \chi(G) + 1$, where $\chi(G)$ is the chromatic number of $G$. Using this lower bound and two classical Ramsey-theoretic results from literature, we answer two questions posed by Milans, Stolee, and West [J. Combinatorics, 2015] (MSW15). First we show that the track number $\tau(L(K_n))$ of the line graph of the complete graphs $K_n$ is at least $\lg\lg n - o(1)$. This is asymptotically tight and it improves the bound of $\Omega(\lg\lg n/ \lg\lg\lg n)$ in MSW15. Next we show that for a family of graphs $\mathcal{G}$, $\{\tau(L(G)):G \in \mathcal{G}\}$ is bounded if and only if $\{\chi(G):G \in \mathcal{G}\}$ is bounded. This affirms a conjecture in MSW15. All our lower bounds apply even if one enlarges the covering family from the family of interval graphs to the family of chordal graphs.