Odd coloring of $k$-trees

TL;DR

This work addresses the odd coloring of $k$-trees, establishing an improved general bound: every $k$-tree is odd $\left(k+2\left\lfloor\log_2 k\right\rfloor+3\right)$-colorable. The authors develop a canonical addition ordering and branch decomposition framework to perform a tight induction on the graph order, yielding a color-extension strategy that carefully preserves the odd condition. They prove tight results for small $k$ by showing 2-trees are odd $4$-colorable and 3-trees are odd $5$-colorable, with the latter outcome supported by extensive case analysis and auxiliary lemmas. The methods provide structural reducibility and color-extension tools for special branches (ears, hats, double hats, and their higher-order analogues), and they connect to known bounds via tree-width considerations, yielding both new bounds and insights for related graph classes.

Abstract

An odd coloring of a graph is a proper coloring such that every non-isolated vertex has a color that appears at an odd number of its neighbors. This notion was introduced by Petrševski and Škrekovski in 2022. In this paper, we focus on odd coloring of $k$-trees, where a $k$-tree is a graph obtained from the complete graph of order $k+1$ by recursively adding a new vertex that is joined to a clique of order $k$ in the former graph. It follows from a result of Cranston, Lafferty, and Song in 2023 that every $k$-tree is odd $(2k+1)$-colorable. We improve this bound to show that every $k$-tree is odd $\left(k+2\left\lfloor\log_2 k\right\rfloor+3\right)$-colorable. Furthermore, when $k=2,3$, we show the tight bound that every 2-tree is odd $4$-colorable and that every 3-tree is odd $5$-colorable.

Figures (10)

• Figure 1: A hat of a 2-tree.
• Figure 2: A double hat of a 2-tree.
• Figure 3: The case where $V(G)\setminus (V_0\cup V_1)=\emptyset$. It follows that $B(\{u,v\},w)\in \mathcal{H}^{(2)}_{0,1,0}(G)$.
• Figure 4: The case where $V(G)\setminus (V_0\cup V_1\cup V_2)=\emptyset$. It follows that $B(\{u',v'\},w')\in \mathcal{H}^{(2)}_{1,0,1}(G)$.
• Figure 6: Case \ref{['case:odd 2tree h101']}

Theorems & Definitions (32)

• Theorem 1: CLS2023
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Conjecture 6
• Lemma 9
• Lemma 10
• proof
• proof