Abstract
In this note, we give a short proof for a theorem on the competition numbers of diamond-free graphs: If a graph G is diamond-free, then the competition number of G is bounded above by \(2 + \tfrac{1}{2}\sum _{v \in V_{\mathrm{ns}}(G)} (\theta _V(N_G(v)) - 2)\), where \(V_{\mathrm{ns}}(G)\) denotes the set of non-simplicial vertices in G and \(\theta _V(N_G(v))\) denotes the minimum number of cliques that cover all the neighbors of a vertex v in G.
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Acknowledgment
The author is grateful to the anonymous reviewers for careful reading and valuable comments. This work was supported by JSPS KAKENHI Grant Number 15K20885.
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Sano, Y. (2016). On the Competition Numbers of Diamond-Free Graphs. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_22
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DOI: https://doi.org/10.1007/978-3-319-48532-4_22
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