Balanced-chromatic number and Hadwiger-like conjectures
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Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the balanced chromatic number, χ_b(Ĝ), of a signed graph Ĝ. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since χ(G)=χ_b(G̃), where G̃ denotes the signed graph obtained from G by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph Ĝ has no negative loop and no K̃_̃t̃-minor, then its balanced chromatic number is at most t-1. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if (G, σ) has no negative loop and no K̃_̃t̃-subdivision, then it admits a balanced 79/2t^2-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions.
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