Line graphs of Multi-Graphs and the forbidden graph $E_6$

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The line graph $\Gamma$ of a multi-graph $\Delta$ is the graph whose vertices are the edges of $\Delta$, where two such edges are adjacent if and only if they meet in a single vertex of $\Delta$. We provide several characterizations of such line graphs and in particular show that a graph is a line graph if and only if it does not contain one of $33$ graphs, all of which correspond to bases of anisotropic vectors of a $6$-dimensional orthogonal geometry of $-$-type over a field with two elements, or, equivalently, to sets of $6$ generating reflections in the Weyl group of type $E_6$.
Original languageEnglish
Article number2105.08618
Number of pages15
Publication statusPublished - 18 May 2021


  • math.CO


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