Balanced-chromatic number and Hadwiger-like conjectures

TL;DR

This paper introduces the concept of balanced chromatic number for signed graphs, proposes a signed version of Hadwiger's conjecture, and establishes its equivalence to the classical conjecture, connecting structural graph theory with signed graph properties.

Contribution

It defines the balanced chromatic number for signed graphs, formulates a signed Hadwiger-like conjecture, and proves its equivalence to the original Hadwiger's conjecture, extending classical results to signed graphs.

Findings

The signed Hadwiger's conjecture is equivalent to the classical Hadwiger's conjecture.

A bound of 79/2 t^2 is established for the balanced chromatic number in graphs with no certain minors.

The work relates subdivisions in signed graphs to their balanced chromatic number.

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 $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ 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 $\tilde{K_t}$-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 $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions.