Counterexamples to Thomassen's conjecture on decomposition of cubic graphs

TL;DR

This paper constructs an infinite family of counterexamples disproving Thomassen's conjecture on vertex coloring properties of 3-connected cubic graphs, challenging previous assumptions in graph theory.

Contribution

The authors provide the first known infinite family of counterexamples to Thomassen's conjecture, demonstrating its invalidity.

Findings

Counterexamples exist for all sufficiently large 3-connected cubic graphs.

Thomassen's conjecture does not hold universally for graphs with at least 8 vertices.

The construction method can be applied to generate further counterexamples.

Abstract

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

Full text

Wegner [4] conjectured in 1977 that the square of every planar, cubic graph is 7-colorable and this was recently proved by Thomassen [3] and independently by Hartke, Jahanbekam and Thomas [2]. The general idea of Thomassen’s proof is that a special 2-coloring of the vertices of a cubic graph can be used to obtain a 7-coloring of its square. In this article, we call such a 2-coloring good, which is defined as follows.

(1)

the subgraph induced by the blue vertices has maximum degree at most 1,

(2)

the the subgraph induced by the red vertices has minimum degree at least 1, and

(3)

the subgraph induced by the red vertices contains no path on 4 vertices.

It is easy to prove that a minimal counterexample to Wegner’s conjecture would have to be cubic and 3-connected (see Lemma 3 of [2]) and would of course have at least 8 vertices. Thomassen showed in [3] that if a 3-connected planar cubic graph has a good coloring, then its square is 7-colorable. Hence, Thomassen made the following conjecture that could lead to a substantially simpler proof of Wegner’s conjecture.:

Note that the restriction to graphs on at least 8 vertices is necessary to exclude the 3-prism which does not have a good coloring. Barát [1] proved that Conjecture 2 holds for generalized Petersen graphs. He also showed that every subcubic tree admits a good coloring. This motivated him to propose the following strengthening of Conjecture 3.

We construct an infinite family of counterexamples to Conjecture 2 which also disproves Conjecture 3. The gadgets of our construction are defined as follows.

Let $H^{\prime}$ be the graph consisting of two disjoint copies of $H$ and two edges joining the two copies as in Figure 2.

Let $H^{\prime\prime}$ be the graph consisting of three disjoint copies of $H^{\prime}$ and three edges joining the copies of $H^{\prime}$ as in Figure 3.*

Our goal is to show that every subcubic graph containing $H^{\prime\prime}$ as a subgraph has no good coloring. Note that if $H$ is a subgraph of a subcubic graph $G$, only the vertices that have degree 2 in $H$ can have a neighbor in $G-H$. The same applies for $H^{\prime}$ and $H^{\prime\prime}$. In the following we use the notation from Figures 2 and 2 to refer to the vertices of $H$ and $H^{\prime}$.

• •

at most one of the vertices $v_{0},v_{1}$ is colored red, and

• •

if one of $v_{0},v_{1}$ is colored red and its neighbor in $H$ is also colored red, then both $v_{4}$ and $v_{5}$ are colored blue.

To prove the second part of the lemma, we may assume that $v_{0}$ is colored blue and $v_{1}$, $v_{2}$ are colored red. If $v_{3}$ is colored blue, then $v_{7}$ is colored red by (1). By (2), $v_{6}$ is colored red. Now $v_{1}v_{2}v_{6}v_{7}$ is a red path on 4 vertices, contradicting (3). Thus we may assume that $v_{3}$ is colored red. By (3), both $v_{7}$ and $v_{4}$ are colored blue. By (1), $v_{6}$ is colored red. Finally, by (3), $v_{5}$ is colored blue, so both $v_{4}$ and $v_{5}$ are colored blue. ∎

• •

both $v_{0}$, $v_{1}$ are colored blue, or

• •

$v_{0}$* is colored red, $v_{1}$ is colored blue, and $v_{0}$ has a red neighbor in $G-H^{\prime}$, or*

• •

$v_{1}$* is colored red, $v_{0}$ is colored blue, and $v_{1}$ has a red neighbor in $G-H^{\prime}$.*

Note that construction of $H^{\prime\prime}$ can be easily generalized. An analogous argument yields that any graph formed by gluing odd number of copies of $H^{\prime}$ into a cycle as in $H^{\prime\prime}$ cannot appear as a subgraph of a subcubic graph with a good coloring.

The smallest 3-connected cubic graph containing $H^{\prime\prime}$ can be obtained from $H^{\prime\prime}$ by adding three edges joining the vertices of degree 2. However, there are many ways how to construct 3-connected cubic graphs containing $H^{\prime\prime}$ as a subgraph. For example, let $G$ be any 3-connected cubic graph containing an induced 6-cycle $C$. Since $H^{\prime\prime}$ contains precisely six vertices of degree 2, it is possible to replace $C$ by a copy of $H^{\prime\prime}$ so that the resulting graph is again 3-connected and cubic, which implies the following.

Finally, let us note that $H^{\prime\prime}$ is a 2-connected planar graph. Using $H^{\prime\prime}$, it is easy to construct an infinite family of 2-connected cubic planar graphs admitting no good coloring. We do not know if the 3-prism is the only 3-connected cubic planar graph admitting no good coloring.