(1,{\lambda})-embedded graphs and the acyclic edge choosability

TL;DR

This paper studies (1,λ)-embedded graphs, showing they are 4-linear and establishing bounds on their acyclic edge choosability for certain surfaces, advancing understanding of graph embedding and coloring.

Contribution

It proves (1,λ)-embedded graphs are 4-linear for all λ and provides new bounds on their acyclic edge choosability for λ=1,2.

Findings

(1,λ)-embedded graphs are 4-linear for all λ.

They are acyclicly edge-(3Δ(G)+70)-choosable for λ=1,2.

Abstract

A (1,{\lambda})-embedded graph is a graph that can be embedded on a surface with Euler characteristic {\lambda} so that each edge is crossed by at most one other edge. A graph G is called {\alpha}-linear if there exists an integral constant {\beta} such that e(G') \leq {\alpha} v(G')+{\beta} for each G'\subseteq G. In this paper, it is shown that every (1,{\lambda})-embedded graph G is 4-linear for all possible {\lambda}, and is acyclicly edge-(3{\Delta}(G)+70)-choosable for {\lambda}=1,2.