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

Licheng Zhang, Zhangdong Ouyang, Yuanqiu Huang

TL;DR

This work investigates claw-free 1-planar graphs and establishes sharp upper bounds on maximum degree and vertex-connectivity. By developing local-structure lemmas for 4-, 6-, and 7-connected 1-planar graphs, the authors prove $\Delta(G)\le 10$ for claw-free 1-planar graphs, refine this to $\Delta(G)\le 8$ for 6-connected cases (sharp), and show $\kappa(G)\le 6$ for claw-free graphs (with 7-connected graphs necessarily containing an induced claw). The results extend Plummer’s claw-free planar graph bounds to the 1-planar setting and include constructions demonstrating sharpness and corollaries about 7-connected graphs not being line graphs. The paper closes with open problems and conjectures regarding potential improvements for higher-connectivity cases, guiding future research in this area.

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.

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

TL;DR

This work investigates claw-free 1-planar graphs and establishes sharp upper bounds on maximum degree and vertex-connectivity. By developing local-structure lemmas for 4-, 6-, and 7-connected 1-planar graphs, the authors prove for claw-free 1-planar graphs, refine this to for 6-connected cases (sharp), and show for claw-free graphs (with 7-connected graphs necessarily containing an induced claw). The results extend Plummer’s claw-free planar graph bounds to the 1-planar setting and include constructions demonstrating sharpness and corollaries about 7-connected graphs not being line graphs. The paper closes with open problems and conjectures regarding potential improvements for higher-connectivity cases, guiding future research in this area.

Abstract

The complete bipartite graph 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.
Paper Structure (7 sections, 11 theorems, 5 equations, 6 figures)

This paper contains 7 sections, 11 theorems, 5 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proofs of main theorems
  4. Proof of Theorem \ref{['main0']}
  5. Proof of Theorem \ref{['thm:main_6connected']}
  6. Proof of Theorem \ref{['thm:connectivity']}
  7. Concluding remarks

Key Result

Theorem 1.1

Let $G$ be a $1$-planar graph. If $G$ is claw-free, then $\Delta(G)\le 10$, and the bound is sharp.

Figures (6)

  • Figure 1: Two 6-connected claw-free 1-planar graphs with maximum degree 8
  • Figure 2: Schematic diagrams for the proofs of Propositions \ref{['prop:4connectedlocal']} and \ref{['prop:6connectedlocal']}
  • Figure 3: A schematic diagram for the proof of Proposition \ref{['prop:7connectedlocal']} (i)
  • Figure 4: Claw-free 1-planar graphs with maximum degree 10
  • Figure 5: (i) A claw-free 1-planar graph $H_0$ with connectivity 3 and maximum degree 10; (ii) a 4-connected claw-free 1-planar graph with maximum degree 8
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Corollary 1.2
  • Lemma 2.1: Erdős Erdos
  • Lemma 2.2: Ouyang, Ge and Chen ouyang2019, Remark 2
  • Lemma 2.3
  • Proposition 2.1
  • Proposition 2.2
  • ...and 1 more