By Ainouche A., Schiermeyer I.

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Extra info for 0-Dual Closures for Several Classes of Graphs

Sample text

Another result of Petersen's fundamental paper [241] is his even factor theorem, that every 2r-regular graph has a 2-factor. The statement that every bridgeless cubic graph has a 1 -factor should perhaps also be formulated as a 2-factor theorem. As pointed out by Hanson, Loten, and Toft [129], every (2r + l)-regular graph with at most 2r bridges has a 2-factor, thus the general Petersen theorem is about 2-factors rather than 1-factors. As we know, every graph G satisfies A{G) < x'(G). In 1916 König [174] proved that equality holds for the class of bipartite graphs, and he deduced as a simple corollary that every regular bipartite graph has a perfect matching.

E^, yt) is a multi-fan at x with respect to e and φ', where a S ^'(z) Π ψ'{ν%), contradicting (a). In order to prove (c), assume that there is a color β £ ^(ί/i) ΓΊ φ^). Since fc > A(G), there is a color α € φ(χ). Then, by (b), there is an (a, /3)-chain with endvertices x and yi as well as an (a, /3)-chain with endvertices x and y^ both with respect to ψ, contradicting yi ψ yj. For the proof of (d), assume that F is maximal. ,

Et} such that then Fl will be extended to the multi-fan Ft+1 = (ei, j / i , . . , e»,i/i,ei+i, j/»+i), where e i + i = e' and j/ i + i is the endvertex of e' distinct from x. This can be done, since we allow a multi-fan to have multiple edges. Then we repeat the subroutine with i replaced by i + 1. If there is no such edge e', then Fl is a maximal multi-fan at x with respect to e and <£, and the algorithm returns the set Z = V(F1). Clearly, the algorithm stops after a finite number of steps, say in step p with F = FP = (ej ] 2/i, - ■ ·, 6p,2/p).