The sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs

TL;DR

This paper establishes sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs, revealing structural limitations and extending known results from planar to 1-planar graphs.

Contribution

It provides the first sharp bounds for maximum degree and vertex-connectivity in claw-free 1-planar graphs, extending prior planar graph results to a broader class.

Findings

Maximum degree of claw-free 1-planar graphs is at most 10.

For 6-connected and optimal 1-planar graphs without induced claws, the maximum degree is at most 8.

Vertex-connectivity of claw-free 1-planar graphs is at most 6.

Abstract

The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend this line of research by establishing some fundamental results for claw-free 1-planar graphs, focusing on upper bounds for maximum degree and vertex-connectivity. We show that the maximum degree of claw-free 1-planar graphs is at most 10, and the bound is sharp. Furthermore, we show that for 6-connected 1-planar graphs and optimal 1-planar graphs under the constraint of forbidding induced claws, the maximum degree has the better upper bound 8. Finally, we show that every 7-connected 1-planar graph contains an induced claw, thereby implying that the vertex-connectivity of claw-free 1-planar graphs is at most 6. For a better comparison, we also refine some known results by Plummer on claw-free planar graphs.