Fractional balanced chromatic number and arboricity of planar (signed) graphs

TL;DR

This work investigates the fractional balanced chromatic number $χ_{fb}(G,σ)$ for planar signed graphs, linking it to the fractional arboricity $a_f(G)$ and addressing longstanding conjectures. Using Wenger-type gadgets, the authors construct planar signed graphs with $χ_{fb} > 2$ and derive a sharp lower bound $χ_{fb} \ge 83/41$ through a bound on missing colors $m_{p,q}$ and iterative gadget composition, while also determining the exact value for the first constructed graph as $χ_{fb}(\widehat{G}_1) = 2 + 2/85$ and proving a limit of $83/41$ for the sequence. In addition, they calculate exact $a_f$ values for the gadget-based graphs, e.g., $a_f(W_1) = 2 + 2/25$ and $a_f$ for a K4-enhanced version as $2 + 2/31$, demonstrating a discrepancy between $χ_{fb}$ and $a_f$ in planar signed graphs. These results collectively advance understanding of fractional colorings in signed planar graphs and provide a framework for asymptotic lower bounds via gadget amplification.

Abstract

A fractional coloring of a signed graph $(G, σ)$ is an assignment of nonnegative weights to the balanced sets (sets which do not induce a negative cycle) such that each vertex has an accumulated weight of at least 1. The minimum total wight among all such colorings is defined to be the fractional balanced chromatic number, denoted by $χ-{fb}(G, σ)$. This value is clearly upper bounded by the fractional arboricity of $G$, denoted $a_f (G)$, where weights are assigned to sets inducing no cycle rather than sets inducing no negative cycle. In this work we present an example of a planar signed simple graph of fractional balanced chromatic number larger than 2, thus in particular refuting a conjecture of Bonamy, Kardos, Kelly, and Postle suggesting that the fractional arboricity of planar graphs is bounded above by 2. By iterating the construction, we show that the supremum of the fractional balanced chromatic number of planar signed simple graphs is at least as $83/41 = 2 + 1/41$. With similar operations, we built a sequence of planar graphs whose limit of fractional arboricity is $a_f (G) = 2 + 2/25$.

Figures (4)

• Figure 1: Completing the positive triangle
• Figure 2: The signed graph $\widehat{W}$
• Figure 3: The signed graph $\widehat{W}'$
• Figure 4: Signed planar graphs satisfying $\chi_{fb}>2$ with an extension

Theorems & Definitions (28)

• Conjecture 1
• Proposition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Remark 5
• Theorem 6
• proof
• Remark 7