Near-perfect matchings in highly connected 1-planar graphs with a local crossing constraint

TL;DR

The paper addresses the existence of near-perfect matchings in highly connected 1-planar graphs under a local crossing constraint, showing that any 6-connected type-2A 1-planar graph has a near-perfect matching and, in fact, satisfies a stronger scattering-number bound $s(G)\le 1$. The approach uses the Tutte-Berge framework together with a reduction that delet es certain crossed edges and contracts components to form a bipartite minor, exploiting the type-2A structure and 6-connectivity to bound $s(G)$. As a corollary, 6-connected type-3 1-planar graphs also have near-perfect matchings; the results illustrate that the 6-connectivity threshold is essential under the local crossing constraint and highlight remaining open questions for broader 1-planar classes. Overall, the work extends matching theory from planar graphs to a carefully constrained 1-planar regime and clarifies the role of local crossing structure in guaranteeing large matchings.

Abstract

For planar graphs, it is well known that high connectivity implies a Hamiltonian cycle and hence any 4-connected planar graph has a near-perfect matching. Nevertheless, whether 6-connected 1-planar graphs admit near-perfect matchings remains largely open. The prior art established this for 4-connected 1-planar graphs only when each crossing involves four endpoints that induce a $K_4$. In this paper, we study 6-connected 1-planar graphs that are drawn such that at all crossings the four endpoints induce a 4-cycle (plus perhaps more edges). We show that these have a near-perfect matching, and in fact even stronger, their scattering number is at most one. Moreover, under the local crossing restriction, the requirement of 6-connectivity is best possible; this is witnessed by explicit constructions due to Biedl and Fabrici et al.

Figures (5)

• Figure 1: A 1-plane graph containing crossings of various types.
• Figure 2: The structure $H$ (with the red and blue cycles highlighted in the electronic version)
• Figure 3: Contraction of an uncrossed edge $uv$ inside a supernode (gray region)
• Figure 4: A vertex-cut $\{a,c,y\}$ arising by the contradiction assumption
• Figure 5: Illustration of Operations I and II.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1
• Lemma 2.1: Tutte-Berge MR0100850
• Lemma 2.2: Karpov zbMATH06347737
• Lemma 2.3
• Remark 2.1
• Lemma 2.4
• proof
• Lemma 2.5
• proof