Vertex Arboricity of Toroidal Graphs with a Forbidden Cycle

TL;DR

This paper proves that toroidal graphs lacking 4-cycles can be partitioned into two forests, establishing a new upper bound on their vertex arboricity and extending known results from planar graphs to toroidal graphs.

Contribution

It confirms that toroidal graphs without 4-cycles have vertex arboricity at most 2, filling a key gap in the understanding of graph arboricity under cycle restrictions.

Findings

Toroidal graphs without 4-cycles have vertex arboricity at most 2.

Extends cycle restriction results from planar to toroidal graphs.

Completes the classification for vertex arboricity of toroidal graphs without certain cycles.

Abstract

The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)\leq 3$. In two recent papers, it was proved that planar graphs without $k$-cycles for some $k\in\{3, 4, 5, 6, 7\}$ have vertex arboricity at most 2. For a toroidal graph $G$, it is known that $a(G)\leq 4$. Let us consider the following question: do toroidal graphs without $k$-cycles have vertex arboricity at most 2? It was known that the question is true for k=3, and recently, Zhang proved the question is true for $k=5$. Since a complete graph on 5 vertices is a toroidal graph without any $k$-cycles for $k\geq 6$ and has vertex arboricity at least three, the only unknown case was k=4. We solve this case in the affirmative; namely, we show that toroidal graphs without 4-cycles have vertex arboricity at most 2.