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

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.

💡 Research Summary

The paper investigates the existence of large matchings in highly connected 1‑planar graphs under a local crossing restriction. A 1‑planar graph can be drawn in the plane so that each edge is crossed at most once. The authors focus on a specific class of such drawings, called type‑2A drawings, where every crossing involves exactly four distinct vertices that form a 4‑cycle (the four endpoints may have additional edges, but the two crossing edges have exactly two associated edges, and those two associated edges form a matching). This local condition prevents complex entanglements of crossings and is a natural relaxation of the stronger “type‑4” condition that characterises locally maximal 1‑planar graphs.

The main result (Theorem 1.1) states that any 6‑connected type‑2A 1‑planar graph possesses a near‑perfect matching, i.e., a matching of size ⌊n/2⌋ where n is the order of the graph. An immediate corollary is that every 6‑connected type‑3 1‑planar graph also has a near‑perfect matching, because a type‑3 drawing is automatically type‑2A. Moreover, the authors prove that the scattering number of such graphs is at most one, which means that for every vertex set S, the number of odd components of G−S never exceeds |S|+1. This is a stronger structural property than merely guaranteeing a large matching.

The proof combines classical matching theory with structural properties of 1‑planar drawings. The authors start from the Tutte‑Berge formula, which expresses the size of a maximum matching α′(G) as

α′(G)=½·min_{S⊆V}(n−odd(G−S)+|S|).

Thus, to obtain α′(G)≥⌊n/2⌋ it suffices to show that for every vertex set S, odd(G−S)−|S|≤1. The 6‑connectivity assumption ensures that any small vertex cut cannot separate the graph into many components, while the type‑2A condition guarantees a “nice” drawing: all associated edges of a crossing are uncrossed. Lemma 2.4 proves that any 4‑connected graph automatically admits a nice drawing, and Lemma 2.7 shows that in a type‑2A drawing, edges belonging to different components of G−S cannot cross each other. Consequently, the number of odd components after removing S is tightly controlled.

A key technical tool is the contraction of uncrossed edges (Lemma 2.3). Because contracting an uncrossed edge preserves 1‑planarity, the authors can iteratively reduce the graph while maintaining both the 6‑connectivity and the type‑2A property. This reduction enables an inductive argument on the number of vertices, ultimately establishing the desired inequality odd(G−S)−|S|≤1 for all S.

The paper also discusses the optimality of the 6‑connectivity requirement. Prior constructions by Biedl and by Fabrici et al. produce 5‑connected type‑3 1‑planar graphs that lack near‑perfect matchings. These examples exploit configurations where some crossings have more than two associated edges (type‑4 or higher), showing that without the 6‑connectivity condition the near‑perfect matching guarantee fails. Hence the result is tight under the given local crossing restriction.

In addition to the matching result, the authors’ analysis yields the bound on the scattering number, a parameter related to graph vulnerability and Hamiltonicity. Since a scattering number ≤1 is a known sufficient condition for Hamiltonicity in many graph classes, the result hints at a possible route toward proving Hamiltonicity for 6‑connected type‑2A 1‑planar graphs, although the paper does not settle that stronger conjecture.

Overall, the contribution is twofold: (1) it introduces a natural local crossing constraint (type‑2A) that is weaker than full maximality yet strong enough to enable powerful combinatorial conclusions; (2) it demonstrates that 6‑connectivity together with this constraint guarantees a near‑perfect matching and a low scattering number, thereby extending classical planar graph results to a broader non‑planar setting. The work opens several avenues for future research, such as relaxing the type‑2A condition, exploring analogous results for lower connectivity combined with degree constraints, or investigating Hamiltonicity under the same framework.