论文标题
伪2因子同构图构的反例的结构
A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture
论文作者
论文摘要
如果$ 2 $ -FACTOR的图形$ g $是\ textit {pseudo $ 2 $ -FACTOR ISOMORPHIC}，如果其所有$ 2 $ factor的周期数的奇偶元是相同的。在[ Abreu，A.A。 Diwan，B。Jackson，D。Labbate和J. Sheehan。伪$ 2 $ -FACTOR同构的常规两部分图。组合理论杂志，第B，98（2）（2008），432-444。]本注释的某些作者给出了伪$ 2 $ 2 $ - 因素异构二键图的部分特征，并推测了$ k_ {3,3,3} $，heatood Graphn and pappus $ 4 $ 4 $ 4 $ 4.在[J。 Goedgebeur。伪$ 2 $因素同构图的反例。委托。 Applied Math., 193 (2015), 57-60.] Jan Goedgebeur computationally found a graph $\mathscr{G}$ on $30$ vertices which is pseudo $2$-factor isomorphic cubic and bipartite, essentially $4$-edge-connected and cyclically $6$-edge-connected, thus refuting the above conjecture.在本说明中，我们描述了如何从Heawood图和广义Petersen Graph $ GP（8,3）$构造的图形，它们分别是Fano $ 7_3 $配置的Levi图和Möbius-Kantor $ 8_3 $配置。 $ \ mathscr {g} $的描述使我们能够同时使用几何学和图理论方法来理解其自动形态群体的订单$ 144 $。此外，我们说明了此图的独特性。
A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the authors of this note gave a partial characterisation of pseudo $2$-factor isomorphic bipartite cubic graphs and conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially $4$-edge-connected ones. In [J. Goedgebeur. A counterexample to the pseudo $2$-factor isomorphic graph conjecture. Discr. Applied Math., 193 (2015), 57-60.] Jan Goedgebeur computationally found a graph $\mathscr{G}$ on $30$ vertices which is pseudo $2$-factor isomorphic cubic and bipartite, essentially $4$-edge-connected and cyclically $6$-edge-connected, thus refuting the above conjecture. In this note, we describe how such a graph can be constructed from the Heawood graph and the generalised Petersen graph $GP(8,3)$, which are the Levi graphs of the Fano $7_3$ configuration and the Möbius-Kantor $8_3$ configuration, respectively. Such a description of $\mathscr{G}$ allows us to understand its automorphism group, which has order $144$, using both a geometrical and a graph theoretical approach simultaneously. Moreover we illustrate the uniqueness of this graph.
