[Paper Review] Tree-chromatic number is not equal to path-chromatic number

Tony Huynh, Ringi Kim|arXiv (Cornell University)|May 22, 2015

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper resolves a long-standing question by proving that the tree-chromatic number and path-chromatic number of a graph are not always equal, introducing a novel graph operation $ R_m(G) $ that increases the path-chromatic number when applied under specific conditions. The key result is an infinite family of $ k $-connected graphs where $ \chi_T(G) \neq \chi_P(G) $, settling a conjecture by Seymour and showing that Mycielski graphs have unbounded path-chromatic number.

ABSTRACT

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $χ(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are different. This settles a question of Seymour. Our results also imply that the path-chromatic numbers of the Mycielski graphs are unbounded.

Motivation & Objective

• To resolve a conjecture by Paul Seymour that the tree-chromatic number and path-chromatic number are not equivalent for all graphs.
• To construct a graph operation $ R_m(G) $ that systematically increases the path-chromatic number under certain conditions.
• To demonstrate that the path-chromatic number of Mycielski graphs is unbounded.
• To establish the existence of an infinite family of $ k $-connected graphs where $ \chi_T(G) \neq \chi_P(G) $.

Proposed method

• Introduces a graph operation $ R_m(G) $ that constructs a new graph from $ G $ using $ m $ copies of $ G $'s vertices and a special vertex $ v_0 $, with adjacency defined by shared index or edge in $ G $.
• Defines a path-decomposition $ P_\sigma^G $ based on vertex enumerations $ \sigma $, where each set $ X_i $ is the closed neighborhood of the first $ i $ vertices minus the first $ i-1 $.
• Introduces the concept of a 'special enumeration' $ \sigma $, where $ \chi(P_\sigma^G) = \chi_P(G) $ and the last vertex in each high-chromatic bag has no earlier neighbors.
• Uses dynamic programming and coloring arguments to analyze chromatic numbers of $ R_m(G) $, particularly focusing on induced subgraphs $ [I, X_j \setminus v_j] $ and their colorability.
• Applies structural lemmas (e.g., Lemma 3.1–3.4) to bound chromatic numbers and derive contradictions when assuming special enumerations exist.
• Proves Theorem 1.3 by showing that $ R_\ell(R_m(G)) $ cannot have a special enumeration, implying its path-chromatic number exceeds $ \chi_P(G) $.

Experimental results

Research questions

• RQ1 Is there a graph for which the tree-chromatic number strictly differs from the path-chromatic number?
• RQ2 Can a graph operation be constructed that increases the path-chromatic number while preserving or altering structural properties?
• RQ3 Do Mycielski graphs have unbounded path-chromatic number?
• RQ4 Is there a function $ f $ such that $ \chi_P(G) \leq f(\chi_T(G)) $ for all graphs $ G $?

Key findings

• The paper constructs an infinite family of $ k $-connected graphs for which $ \chi_T(G) \neq \chi_P(G) $, proving that the two parameters are not equivalent.
• For any graph $ G $ with $ \chi_P(G) = k $, the operation $ R_m(G) $ increases the path-chromatic number to $ k+1 $ if $ G $ has no special enumeration, otherwise it remains $ k $.
• Applying $ R_m $ twice, i.e., $ R_\ell(R_m(G)) $ with $ \ell \geq m(n+1)+k+3 $, ensures $ \chi_P(R_\ell(R_m(G))) > k $, proving strict increase.
• The path-chromatic number of Mycielski graphs is unbounded, as their construction allows repeated application of $ R_m $.
• The existence of graphs with $ \chi_T(G) < \chi_P(G) $ implies that no function $ f $ with $ \chi_P(G) \leq f(\chi_T(G)) $ can exist universally.
• The proof relies on structural coloring arguments and contradiction via the non-existence of special enumerations in $ R_\ell(R_m(G)) $.