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Brushing Number and Zero-Forcing Number of Graphs and their Line Graphs

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arxiv 1609.05854 v1 pith:O3U45HVW submitted 2016-09-19 math.CO

Aras Erzurumluoglu , David Pike , Karen Meagher This is my paper
classification math.CO
keywords numbergraphlinezero-forcingbrushingprovegraphsbound
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In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.

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