arXiv subjects

Shohei Koizumi

Publications and source records attributed to Shohei Koizumi.

4 recordsLinked to original sources

arXiv2026

Linkage problem on optimal $1$-planar graphs

Enami and Maezawa give a complete characterization of $(s_1, s_2, \ldots, s_k)$-linked planar graphs for any $k$-tuple of positive integers. In this paper, we investigate linkage problems for optimal 1-planar graphs. In particular, we show that every optimal 1-planar graph with connectivity $6$ is $(5, 5)$-linked. Moreover, for an optimal $1$-planar graph $G$ that is not $(2,2,1)$-linked, we characterize disjoint vertex subsets $S_1, S_2, S_3$ in $G$ with $|S_1|=|S_2|=2$ and $|S_3|=1$ such that $G$ is not $\{S_1,S_2,S_3\}$-linked.

math.CO↗

arXiv2026

Another proof of the result on rotation compatible planar covers

Negami's Planar Cover Conjecture asserts that a connected graph has a finite planar cover if and only if it can be embedded on the projective plane. While this statement has already been proven for rotation compatible planar covers, namely covers equipped with a certain condition on the rotation system, the existing proof relies on advanced algebraic and topological methods. In this paper, we provide another proof of this result, focusing primarily on combinatorial arguments based on a structural analysis with respect to a spanning tree in the base graph.

math.CO↗

arXiv2025

The matching extendability of optimal $1$-embedded graphs on the projective plane

In this paper, we discuss matching extendability of optimal $1$-projective plane graphs (abbreviated as O1PPG), which are drawn on the projective plane $P^2$ so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n- 4$ edges. We first show that every O1PPG of even order is $1$-extendable. Next, we characterize $2$-extendable O1PPG's in terms of a separating cycle consisting of only non-crossing edges. Moreover, we characterize O1PPG's having connectivity exactly $5$. Using the characterization, we further identify three independent edges in those graphs that are not extendable.

math.CO↗

arXiv2025

Connectivity and matching extendability of optimal $1$-embedded graphs on the torus

In this paper, we discuss optimal $1$-toroidal graphs (abbreviated as O1TG), which are drawn on the torus so that every edge crosses another edge at most once, and has $n$ vertices and exactly $4n$ edges. We first consider connectivity of O1TGs, and give the characterization of O1TGs having connectivity exactly $k$ for each $k\in \{4, 5, 6, 8\}$. In our argument, we also show that there exists no O1TG having connectivity exactly $7$. Furthermore, using the result above, we discuss extendability of matchings, and give the characterization of $1$-, $2$- and $3$-extendable O1TGs in turn.

math.CO↗