[Paper Review] A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number

Alvaro Carbonero, Patrick Hompe+2 more|arXiv (Cornell University)|Jan 20, 2022

Advanced Graph Theory Research9 references14 citations

TL;DR

This paper constructs a family of graphs with arbitrarily large chromatic number and clique number at most 3, such that all triangle-free induced subgraphs have chromatic number at most 4. The construction uses a directed graph with large dichromatic number, no induced odd directed cycles of length at least 5, and bounded clique number, thereby disproving a long-standing conjecture about induced subgraphs of graphs with high chromatic number and small clique number.

ABSTRACT

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.

Motivation & Objective

• To disprove a well-known conjecture stating that every graph with high chromatic number and bounded clique number must contain an induced subgraph with high chromatic number and no triangles.
• To investigate the structural limitations of induced subgraphs in graphs with high chromatic number and small clique number.
• To explore the dichromatic number behavior in directed graphs with bounded clique number and no induced odd directed cycles of length at least 5.
• To determine whether there exists a function bounding the dichromatic number in terms of clique number for digraphs excluding induced odd directed cycles of length at least 5.

Proposed method

• Construct a sequence of directed graphs {Dn} recursively, starting from a single vertex and iteratively combining n−1 copies of Dn−1 with new vertices indexed by tuples of vertices from the copies.
• Define D′n as a transformation of Dn where edges are added between pairs of vertices based on the length modulo 3 of the unique directed path between them in Dn.
• Partition edges in D′n into positive and negative signs based on path length modulo 3: length ≡1 mod 3 gives positive edges, length ≡2 mod 3 gives reverse edges.
• Prove that Dn is acyclic and has at most one directed path between any two vertices, ensuring D′n is a well-defined simple digraph.
• Use the sign structure of edges in D′n to show that no three vertices can all be pairwise connected, preventing triangles in the underlying undirected graph.
• Apply a 4-coloring argument based on vertex partitioning into sets avoiding positive or negative edges to bound the chromatic number of any triangle-free induced subgraph.

Experimental results

Research questions

• RQ1 Does every graph with high chromatic number and clique number at most 3 contain an induced subgraph with high chromatic number and no triangles?
• RQ2 Is there a function f such that the dichromatic number of a digraph is bounded by f(ω(D)) whenever the digraph has no induced odd directed cycle of length at least 5?
• RQ3 Can the dichromatic number be bounded in terms of clique number for digraphs excluding induced directed cycles of length not equal to l, for some l?
• RQ4 Is there a function f such that the dichromatic number of a digraph is bounded by f(ω(D)) when the digraph has no induced directed cycles of length at least 4?

Key findings

• For every n, there exists a graph G with χ(G) ≥ n and ω(G) ≤ 3 such that every induced subgraph H with ω(H) ≤ 2 satisfies χ(H) ≤ 4, thus disproving Conjecture 1.1 for all r ≥ 5.
• The underlying undirected graph of D′n has chromatic number at least n and clique number at most 3, confirming the existence of such graphs for arbitrarily large chromatic number.
• The digraph D′n has no induced directed cycles of odd length at least 5, and its underlying undirected graph has clique number at most 3.
• The dichromatic number of D′n is at least n, and D′n has no induced odd directed cycles of length at least 5, showing that the dichromatic number is not bounded by a function of the clique number in this class.
• The chromatic number of any triangle-free induced subgraph of the underlying undirected graph of D′n is at most 4, due to a 4-coloring argument based on edge sign partitioning.
• The construction provides a counterexample to the conjecture that all graphs with high chromatic number and small clique number must contain an induced subgraph with high chromatic number and no triangles, leaving only the case r = 4 open.