1. The maximum matching extendability and factor-criticality of 1-planar graphs
- Author
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Zhang, Jiangyue, Wu, Yan, and Zhang, Heping
- Subjects
Mathematics - Combinatorics ,05C70, 05C10 - Abstract
A graph is $1$-$planar$ if it can be drawn in the plane so that each edge is crossed by at most one other edge. Moreover, a 1-planar graph $G$ is $optimal$ if it satisfies $|E(G)|=4|V(G)|-8$. J. Fujisawa et al. [16] first considered matching extension of optimal 1-planar graphs, obtained that each optimal 1-planar graph of even order is 1-extendable and characterized 2-extendable optimal 1-planar graphs and 3-matchings extendable to perfect matchings as well. In this short paper, we prove that no optimal $1$-planar graph is 3-extendable. Further we mainly obtain that no 1-planar graph is 5-extendable by the discharge method and also construct a 4-extendable 1-planar graph. Finally we get that no 1-planar graph is 7-factor-critical and no optimal 1-planar graph is 6-factor-critical., Comment: 13 pages, 8 figures
- Published
- 2022