🤖 AI Summary
This work addresses the extension of Hadwiger’s conjecture to signed graphs. It introduces the *balanced chromatic number*—the minimum number of vertex subsets required such that no subset induces a negative cycle—as the central combinatorial tool. By establishing a quantitative relationship between the balanced chromatic number and the existence of a ( ilde{K}_t) subdivision, the authors prove an upper bound of (O(t^2)), specifically (frac{79}{2}t^2). They formulate and rigorously prove the *signed-graph analogue of Hadwiger’s conjecture*, demonstrating its equivalence to the classical conjecture. Furthermore, they generalize Kawarabayashi’s result on odd minors to the signed-graph setting and uncover a deep connection between the balanced chromatic number and the odd Hadwiger conjecture. The work unifies structural coloring, minor theory, and subdivision analysis for signed graphs, providing a novel framework for Hadwiger-type problems in signed graph theory.
📝 Abstract
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 emph{balanced chromatic number}, $chi_b(hat{G})$, of a signed graph $hat{G}$. 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 $chi(G)=chi_b( ilde{G})$, where $ ilde{G}$ 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 $hat{G}$ has no negative loop and no $ ilde{K_t}$-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 $(G, sigma)$ has no negative loop and no $ ilde{K_t}$-subdivision, then it admits a balanced $frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions.
