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Brooks' theorem for signed graphs with $Δ=3$

Reza Naserasr, Huan Zhou

TL;DR

This work extends Brooks-type ideas to signed graphs by establishing a sharp upper bound on the circular chromatic number for connected signed graphs with maximum degree $3$, namely $\chi_c(G,\sigma) \leq \frac{10}{3}$ when the graph is not switching-isomorphic to $(K_{4},-)$. The authors develop a $10/3$-coloring framework via edge-sign preserving homomorphisms to $K_{p;q}^s$ and pivotal reductions to $\widehat{K}_{10;3}^s$, complemented by a list-coloring approach and structural analysis. A potential function and a comprehensive discharging argument yield forbidden configurations for any potential counterexample, proving the main theorem and its tightness (e.g., on the Petersen graph with an appropriate signature). The results illuminate the odd-degree case for signed graphs and provide a blueprint for extending similar bounds to higher degrees.

Abstract

Circular $r$-coloring of a signed graph $(G,σ)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a positive connection, the distance of one from the antipodal of the other is at least $1$. A signed graph $(G,σ)$ admits a circular $r$-coloring for some values of $r$ if and only if it has no negative loop. The smallest value of such $r$ is the circular chromatic number, denoted $χ_{c}(G,σ)$. The circular chromatic number is a refinement of the balanced chromatic number, which is mostly studied under the equivalent term $0$-free coloring in the literature. Extending Brooks' theorem, Má\v cajová, Raspaud, and Škoviera showed that if $Δ(G)$ is an even number, $G$ is connected, and $(G,σ)$ is not (switching) isomorphic to $(K_{Δ+1},-)$ or $C_{-\ell}$ (when $Δ(G)=2$), then $χ_c(G,σ)\leq Δ(G)$ and that the upper bound is tight. For the odd values of $Δ(G)$, assuming a connected signed graph $(G,σ)$ is not isomorphic to $(K_{Δ+1},-)$, determining the best upper bound for $χ_c(G, σ)$ proves to be more of a challenge. In this work, addressing the first step of this question, we show that if $(G, σ)$ is a signed graph of maximum degree 3 with no component isomorphic to $(K_4, -)$, then $χ_{c}(G, σ)\leq \frac{10}{3}$. The upper bound is tight even among signed cubic graphs of girth 5. In particular, there is a signature on the Petersen graph for which the upper of $\frac{10}{3}$ is achieved.

Brooks' theorem for signed graphs with $Δ=3$

TL;DR

This work extends Brooks-type ideas to signed graphs by establishing a sharp upper bound on the circular chromatic number for connected signed graphs with maximum degree , namely when the graph is not switching-isomorphic to . The authors develop a -coloring framework via edge-sign preserving homomorphisms to and pivotal reductions to , complemented by a list-coloring approach and structural analysis. A potential function and a comprehensive discharging argument yield forbidden configurations for any potential counterexample, proving the main theorem and its tightness (e.g., on the Petersen graph with an appropriate signature). The results illuminate the odd-degree case for signed graphs and provide a blueprint for extending similar bounds to higher degrees.

Abstract

Circular -coloring of a signed graph is a mapping of its vertices to a circle of circumference such that: I. each pair of vertices with a negative connection is at distance at least , and II. for each pair with a positive connection, the distance of one from the antipodal of the other is at least . A signed graph admits a circular -coloring for some values of if and only if it has no negative loop. The smallest value of such is the circular chromatic number, denoted . The circular chromatic number is a refinement of the balanced chromatic number, which is mostly studied under the equivalent term -free coloring in the literature. Extending Brooks' theorem, Má\v cajová, Raspaud, and Škoviera showed that if is an even number, is connected, and is not (switching) isomorphic to or (when ), then and that the upper bound is tight. For the odd values of , assuming a connected signed graph is not isomorphic to , determining the best upper bound for proves to be more of a challenge. In this work, addressing the first step of this question, we show that if is a signed graph of maximum degree 3 with no component isomorphic to , then . The upper bound is tight even among signed cubic graphs of girth 5. In particular, there is a signature on the Petersen graph for which the upper of is achieved.
Paper Structure (8 sections, 27 theorems, 4 equations, 12 figures)

This paper contains 8 sections, 27 theorems, 4 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Brooks type theorems
  3. Critical signed graphs
  4. The 10/3-circular coloring and homomorphism
  5. List $10/3$-coloring
  6. Structural properties
  7. Discharging
  8. Conclusion

Key Result

Theorem 2

If $(G, \sigma)$ is a connected signed simple graph of maximum degree at most $p$ where $p$ is a positive even integer, $p\geq 4$, unless $(G, \sigma)$ is (switching) isomorphic to $(K_{p+1}, -)$, it admits a circular $p$-coloring. Moreover, the upper bound of $p$ is tight.

Figures (12)

  • Figure 1: Circular clique structures
  • Figure 2: The signed graphs $\widehat{T}$ and $\widehat{T}^+$
  • Figure 6: The signed graph on the cube
  • Figure 7: The signed Petersen graph
  • Figure 8: The indicator construction $(I, s, t)$
  • ...and 7 more figures

Theorems & Definitions (29)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Corollary 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 19 more