Decompositions of graphs of nonnegative characteristic with some forbidden subgraphs

TL;DR

This paper studies graph decompositions into acyclic orientations and low-degree subgraphs for graphs on surfaces, establishing new conditions under which such decompositions exist, generalizing previous results.

Contribution

It introduces new decomposability results for graphs on surfaces with forbidden cycles, extending prior work with broader conditions.

Findings

Graphs with no certain chord cycles are (3,1)-decomposable.

Graphs with no specific cycles are (2,1)-decomposable.

Results generalize previous theorems on graph decompositions.

Abstract

A {\em $(d,h)$-decomposition} of a graph $G$ is an order pair $(D,H)$ such that $H$ is a subgraph of $G$ where $H$ has the maximum degree at most $h$ and $D$ is an acyclic orientation of $G-E(H)$ of maximum out-degree at most $d$. A graph $G$ is {\em $(d, h)$-decomposable} if $G$ has a $(d,h)$-decomposition. Let $G$ be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove the following results. (1) If $G$ has no chord $5$-cycles or no chord $6$-cycles or no chord $7$-cycles and no adjacent $4$-cycles, then $G$ is $(3,1)$-decomposable, which generalizes the results of Chen, Zhu and Wang [Comput. Math. Appl, 56 (2008) 2073--2078] and the results of Zhang [Comment. Math. Univ. Carolin, 54(3) (2013) 339--344]. (2) If $G$ has no $i$-cycles nor $j$-cycles for any subset $\{i,j\}\subseteq \{3,4,6\}$ is $(2,1)$-decomposable, which generalizes the results of Dong and Xu [Discrete Math. Alg. and Appl., 1(2) (2009), 291--297].