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Abstract
The track number of a graph is the minimum number of interval graphs whose union is . We show that the track number of the line graph of a triangle-free graph is at least , where is the chromatic number of . 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 of the line graph of the complete graphs is at least . This is asymptotically tight and it improves the bound of in MSW15. Next we show that for a family of graphs , is bounded if and only if is bounded. This affirms a conjecture in MSW15. All our lower bounds apply…
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