arXiv:0908.1506v2 [math.CO] 10 Aug 2013 orientation M acig eta cycle central a matching, in set Let Introduction 1 [email protected] [email protected] email: USA; aewdho fffinBae n oye rpson Graphs Polyhex and Braces Pfaffian of Face-width ∗ † M uhthat such colo ahmtc n ttsis azo nvriy Lanzh University, Lanzhou Statistics, and Mathematics of Un School State Tennessee Middle Sciences, Mathematical of Department M G yl of cycle A . k nyi ti smrhct h ue h ewo rp or graph Heawood the gr cube, polyhex the to bipartite isomorphic a is that it show if C we only 2, graphs. face-width polyhex with Pfaffian p furt characterizing graphs with obtain in we surface brace, useful a cubic are on Pfaffian which a brace For Pfaffian 3. most a at of Combin. face-width M embedding J. every [Ann. that [Electorn. Thomas show McCuaig and independently Seymour applyin and Robertson, by to paper, 929-975], match this due In perfect braces of Pfaffian time. number of polynomial The in computed cycle. be the can of direction either in C eagahwt etxset vertex with graph a be fidpnetegssc hteeyvre of vertex every that such edges independent of ≥ see and even is graph A ,adalnnbprieplhxgah r Pfaffian. are graphs polyhex non-bipartite all and 6, D C to is G G G G M nee cycle even an , sPaa fi a noinainsc htec eta cycl central each that such orientation an has it if Pfaffian is − atraig(..teegsof edges the (i.e. -alternating is V central ( C C a efc acig a nodnme fegsdirected edges of number odd an has matching) perfect a has ) utb fee ie nohrwords, other In size; even of be must ogYe Dong if G V C ( − G ∗ is Surfaces n deset edge and ) V n eigZhang Heping and Abstract dl oriented oddly ( C a efc acig If matching. perfect a has ) 1 C E G ( if lent nadoff and on alternate G sicdn iheatyoeedge one exactly with incident is C .A ). C u as 300 hn;email: China; 730000, Gansu ou, vriy ufesoo N37132, TN Murfreesboro, iversity, a nodnme fedges of number odd an has † k e tutr properties structure her miigwt polyhex with ombining nso fffingraphs Pfaffian of ings × efc matching perfect h characterization the g p sPaa fand if Pfaffian is aph 1(04 R9,we #R79], (2004) 11 G K a efc matching perfect a has 2 stv eu has genus ositive o vnintegers even for t.10(1999) 150 ath. G e a perfect a has M C .Fran For ). (i.e. of G sa is directed in either direction of the cycle. An orientation of G is Pfaffian if every central cycle of G is oddly oriented. A graph G is Pfaffian if it has a Pfaffian orientation. It is known that if G has a Pfaffian orientation D, then the number of of G can be obtained by computing the determinant of the skew adjacency matrix of D [13]. The Pfaffian orientation was first introduced by Kasteleyn [6] for solving 2-dimensional Ising problem. It is important to determine whether a given graph is Pfaffian or not. Kasteleyn [6] showed that every admits a Pfaffian orientation. Little [11] obtained a characterization for Pfaffian , that a bipartite graph is Pfaffian if only if it has no matching minor isomorphic to K3,3. The problem characterizing Pfaffian bipartite graph is related to many interesting problems, such as the P´olya permanent problem, the sign-nonsingular matrix problem, etc. But the problem of characterizing Pfaffian non-bipartite graphs remains open. Readers may refer to a survey of Thomas [24] on the Pfaffian orientations of graphs. A connected graph G is k-extendable (|V (G)| ≥ 2k + 2) if G has k independent edges and each set of k independent edges G is contained in a perfect matching. A k-extendable graph is (k − 1)-extendable and (k + 1)-connected [19, 13]. A 2-extendable bipartite graph is also called a brace. So a brace is 3-connected. A 3-connected bicritical graph G (the deletion of any pair of distinct vertices of G results in graph with a perfect matching) is called a brick. Every bicritical graph is 1-extendable and non-bipa