Extension of the Gyárfás-Sumner conjecture to signed graphs

TL;DR

This work extends the Gyárfás-Sumner paradigm to signed graphs by studying the balanced chromatic number $\chi_b$ within hereditary classes. It establishes a structural characterization for GS sets of order 2: the first forbidden must be switching-equivalent to $\widehat{(K_3,-)}$ or $\widehat{(K_4,-)}$, and the second must be a linear forest, with precise bounds in each case. It proves that $\{\widehat{(K_3,-)},F\}$ is GS for every linear forest $F$, and that $\{\widehat{(K_4,-)},F\}$ is GS when each component of $F$ has length at most 4 (with conjecture for all linear forests). Additionally, the paper develops two unbounded-\chi_b constructions via signed shift graphs and signed line graphs to illustrate limitations of GS-sets and provides corollaries such as the GS-status of $\mathrm{Forb}_{ind}\{\widehat{(K_4,-)},P_4\}$, including a 6-color bound on the negative-edge subgraph in that class. Overall, the work lays groundwork for a signed-graph analogue of Gyárfás-Sumner, linking switching, balance, and forbiddances in a rigorous framework and suggesting directions for higher-order GS-sets and neighborhood structure analyses.

Abstract

The balanced chromatic number of a signed graph G is the minimum number of balanced sets that cover all vertices of G. Studying structural conditions which imply bounds on the balanced chromatic number of signed graphs is among the most fundamental problems in graph theory. In this work, we initiate the study of coloring hereditary classes of signed graphs. More precisely, we say that a set F = {F_1, F_2, ..., F_l} is a GS (for Gyárfás-Sumner) set if there exists a constant c such that signed graphs with no induced subgraph switching equivalent to a member of F admit a balanced c-coloring. The focus of this work is to study GS sets of order 2. We show that if F is a GS set of order 2, then F_1 is either (K_3, -) or (K_4, -), and F_2 is a linear forest. In the case of F_1 = (K_3, -), we show that any choice of a linear forest for F_2 works. In the case of F_1 = (K_4, -), we show that if each connected component of F_2 is a path of length at most 4, then {F_1, F_2} is a GS set.

Figures (9)

• Figure 1: $(Q_3,+) \simeq (Q_3,\sigma) \simeq (Q_3,-)$
• Figure 2: A graph in $\mathop{\mathrm{Forb}}\nolimits_{ind}{\{}P_3,(K_3,+),\widehat{(K_4,-)}\}$
• Figure 3: The biggest connected component of the signed shift graph $\widehat{S}_{3,6}$
• Figure 4: Illustration of the proof, with $F_1=F_2=P_4$
• Figure 5: $(K_3,-)$ and $(K_4,M)$

Theorems & Definitions (48)

• theorem 1
• definition 2
• lemma 3
• lemma 4
• definition 5
• lemma 6
• proof
• corollary 7
• definition 8
• definition 9