曲面上Pfrongian支撑和Polyhex图的面宽
Face-width of Pfaffian Braces and Polyhex Graphs on Surfaces
摘要：
如果图$G$具有一个方向使得每个中心循环$C$（即$C$是偶数并且$G-V(C)$具有完美匹配）具有指向循环的任一方向的奇数条边，则图$G$是Pfuman的。Pfronian图的完美匹配数可以在多项式时间内计算。本文应用Robertson，Seymour和Thomas [Ann.Math.150（1999）929-975]，McCuaig [Electorn. J. Combin。11（2004）#R79]，我们证明了Pfwean支撑在具有正亏格的曲面上的每个嵌入的面宽至多为3。对于Pfwean三次支撑，我们得到了进一步的结构性质，这些性质对于刻画Pfwean多六角图是有用的.结合面宽为2的polyhex图，证明了二部polyhex图是Pfuman的当且仅当它同构于立方体，Heawood图或$C_k\times K_2$（对于偶数$k\ge 6$），所有非二部polyhex图都是Pfuman的.
英文摘要：
A graph $G$ is Pfaffian if it has an orientation such that each central cycle $C$ (i.e. $C$ is even and $G-V(C)$ has a perfect matching) has an odd number of edges directed in either direction of the cycle. The number of perfect matchings of Pfaffian graphs can be computed in polynomial time. In this paper, by applying the characterization of Pfaffian braces due to Robertson, Seymour and Thomas [Ann. Math. 150 (1999) 929-975], and independently McCuaig [Electorn. J. Combin. 11 (2004) #R79], we show that every embedding of a Pfaffian brace on a surface with positive genus has face-width at most 3. For a Pfaffian cubic brace, we obtain further structure properties which are useful in characterizing Pfaffian polyhex graphs. Combining with polyhex graphs with face-width 2, we show that a bipartite polyhex graph is Pfaffian if and only if it is isomorphic to the cube, the Heawood graph or $C_k\times K_2$ for even integers $k\ge 6$, and all non-bipartite polyhex graphs are Pfaffian.
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