[Paper Review] On Sylvester Colorings of Cubic Graphs

Anush Hakobyan, Vahan Mkrtchyan|arXiv (Cornell University)|Nov 8, 2015

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper proves the Sylvester coloring conjecture for cubic pseudo-graphs and establishes two key results: (1) every cubic graph admits an S-coloring where at least 4/5 of its vertices satisfy the Sylvester condition, and (2) every claw-free cubic graph admits a full S-coloring. These results advance the understanding of edge-coloring in cubic graphs and support the broader conjecture that the Sylvester graph S on 10 vertices is a universal colorer for all cubic graphs.

ABSTRACT

If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a proper edge-coloring $f$ with edges of $H$, such that for each vertex $x$ of $G$, there is a vertex $y$ of $H$ with $f(\partial_G(x))=\partial_H(y)$. If $G$ admits an $H$-coloring, then we will write $H\prec G$. The Petersen coloring conjecture of Jaeger states that for any bridgeless cubic graph $G$, one has: $P\prec G$. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph $G$ one has: $S\prec G$. Here $S$ is the Sylvester graph on $10$ vertices. In this paper, we prove the analogue of Sylvester coloring conjecture for cubic pseudo-graphs. Moreover, we show that if $G$ is any connected simple cubic graph $G$ with $G\prec P$, then $G = P$. This implies that the Petersen graph does not admit an $S_{16}$-coloring, where $S_{16}$ is the smallest connected simple cubic graph without a perfect matching. $S_{16}$ has $16$ vertices. %We conjecture that there are infinitely many connected cubic simple graphs which do not admit an %$S_{16}$-coloring. Finally, we obtain $2$ results towards the Sylvester coloring conjecture. The first result states that any cubic graph $G$ has a coloring with edges of Sylvester graph $S$ such that at least $\frac45$ of vertices of $G$ meet the conditions of Sylvester coloring conjecture. The second result states that any claw-free cubic graph graph admits an $S$-coloring. This results is an application of our result on cubic pseudo-graphs.

Motivation & Objective

• To prove the Sylvester coloring conjecture for cubic pseudo-graphs.
• To establish a 4/5 vertex satisfaction bound for S-coloring in general cubic graphs.
• To prove that all claw-free cubic graphs admit a full S-coloring.
• To demonstrate that the Petersen graph P is the only graph satisfying P ≺ G among connected simple cubic graphs.
• To show that S₁₆, the smallest 16-vertex cubic graph without a perfect matching, does not admit an S-coloring.

Proposed method

• Use of edge-coloring mappings f: E(G) → E(H) such that for each vertex x in G, the incident edges under f match the edge set of some vertex in H.
• Construction of S-colorings via restriction and extension techniques on subgraphs and pseudo-graphs.
• Application of the transitive property of the ≺ relation to chain colorings across graphs.
• Use of structural decomposition: replacing vertices with triangles to relate H to G via pseudo-graphs.
• Induction on |V(G)| for claw-free graphs, leveraging known characterizations of such graphs.
• Leveraging the fact that S₁₂ can be obtained from S₄ by vertex-triangle replacement, and using S₄-colorability of pseudo-graphs to extend to S₁₂ and then to S.

Experimental results

Research questions

• RQ1 Does every cubic pseudo-graph admit an S-coloring, as claimed by the Sylvester coloring conjecture?
• RQ2 Can a coloring be constructed such that at least 4/5 of the vertices in any cubic graph satisfy the Sylvester condition?
• RQ3 Does every claw-free cubic graph admit a full S-coloring?
• RQ4 Is the Petersen graph the only connected simple cubic graph G for which P ≺ G holds?
• RQ5 Does the 16-vertex cubic graph S₁₆, lacking a perfect matching, admit an S-coloring?

Key findings

• The Sylvester coloring conjecture holds for all cubic pseudo-graphs.
• For any cubic graph G, there exists an S-coloring such that at least 4/5 of the vertices satisfy the Sylvester condition, i.e., |V(f)| ≥ (4/5)|V(G)|.
• Any claw-free cubic graph admits a full S-coloring, i.e., S ≺ G.
• The Petersen graph P is the only connected simple cubic graph satisfying P ≺ G.
• The 16-vertex cubic graph S₁₆ does not admit an S-coloring, as it lacks a perfect matching and is not S-colorable.