AI 中文总结
本文研究最优2-平面图的匹配可扩性，证明偶数阶4-连通最优2-平面图是1-可扩的，不存在5-可扩的最优2-平面图，构造了4-可扩的最优2-平面图，还证明6-连通最优2-平面图满足距离3的m-可扩性。
AI 中文摘要
若图G可在平面上绘制，使得每条边最多被两条其他边交叉，则称G为2-平面图。已知对于2-平面图G，其边数满足|E(G)|≤5|V(G)|−10，当等号成立时，称G为最优2-平面图。本文研究最优2-平面图的匹配可扩性，利用局部最优性证明：任意偶数阶的4-连通最优2-平面图G是1-可扩的，并给出G为2-可扩的判定准则；还证明不存在5-可扩的最优2-平面图，且基于正十二面体构造了一个4-可扩的最优2-平面图；最后证明，对任意m≥0，任意至少含2m+2个顶点的偶数阶6-连通最优2-平面图G是距离3的m-可扩的。
英文摘要
A graph is 2-planar if it can be drawn in the plane such that each edge is crossed by at most two other edges. It is known that for a 2-planar graph $G$, $|E(G)| \le 5|V(G)| - 10$. When the equality holds, we call $G$ an optimal 2-planar graph. This paper investigates the matching extendability of optimal 2-planar graphs. By local optimality, we prove that every 4-connected optimal 2-planar graph $G$ of even order is 1-extendable, and give a criterion for $G$ to be 2-extendable. We also prove that no optimal 2-planar graph is 5-extendable and construct a 4-extendable optimal 2-planar graph based on the dodecahedron. Finally, we show that every 6-connected optimal 2-planar graph of even order with at least $2m+2$ vertices is distance 3 $m$-extendable for any $m \ge 0$.
Comments19 pages, 12 figures
