Some New Characterizations of Graph Colorability and of Blocking Sets of Projective Spaces
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Let $G=(V,E)$ be a graph and $q$ be an odd prime power. We prove that $G$ possess a proper vertex coloring with $q$ colors if and only if there exists an odd vertex labeling $x\in F_q^V$ of $G$. Here, $x$ is called odd if there is an odd number of partitions $\pi=\{V_1,V_2,\dotsc,V_t\}$ of $V$ whose blocks $V_i$ are \(G\)-bipartite and \(x\)-balanced, i.e., such that $G|_{V_i}$ is connected and bipartite, and $\sum_{v\in V_i}x_v=0$. Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game.
2017 ◽
Vol 09
(01)
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pp. 1750014
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1980 ◽
Vol 32
(3)
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pp. 628-630
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2018 ◽
Vol 10
(01)
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pp. 1850014
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2012 ◽
Vol 49
(2)
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pp. 156-169
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2018 ◽
Vol 7
(4.10)
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pp. 393
2018 ◽
Vol 2
(1)
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pp. 30
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