Download PDF by Professor Cun-Quan Zhang: Circuit Double Cover of Graphs

By Professor Cun-Quan Zhang

ISBN-10: 0521017564

ISBN-13: 9780521017565

ISBN-10: 0521528844

ISBN-13: 9780521528849

The recognized Circuit Double conceal conjecture (and its a number of variations) is taken into account one of many significant open difficulties in graph idea due to its shut courting with topological graph idea, integer move idea, graph coloring and the constitution of snarks. you will kingdom: each 2-connected graph has a relatives of circuits overlaying each part accurately two times. C.-Q. Zhang presents an updated evaluation of the topic containing all the thoughts, tools and effects constructed to assist clear up the conjecture because the first booklet of the topic within the Nineteen Forties. it's a important survey for researchers already engaged on the matter and a becoming advent for these simply coming into the sector. The end-of-chapter routines were designed to problem readers at each point and tricks are supplied in an appendix.

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1 Faithful circuit cover The concept of faithful circuit cover is not only a generalization of the circuit double cover problem, but also an inductive approach to the CDC conjecture in a very natural way. Let Z+ be the set of all positive integers, and Z be the set of all non-negative integers. 1 Let G be a graph and w : E(G) → Z+ . A family F of circuits (or even subgraphs) of G is a faithful circuit cover (or faithful even subgraph cover, respectively) with respect to w if each edge e is contained in precisely w(e) members of F.

Let {C1 , . . , Ct } (⊆ F) be a circuit chain joining the endvertices x0 and y0 of e0 . If F − {C1 , . . , Ct } = ∅, then every member of F − {C1 , . . , Ct }) is a removable circuit of (G, w). This contradicts that (G, w), as a minimal contra pair, has no removable circuit. So, let F = {C1 , . . , Ct } be a circuit chain joining x0 and y0 where Cα ∩Cβ = ∅ if and only if α = β ±1. We also notice that Ew=2 consists of edges of {e0 } t−1 i=1 [E(Ci )∩E(Ci+1 )] and is, therefore, a perfect matching of the cubic graph G.

3), Si be the spanning even subgraph of Gi obtained from S by shrinking all components of S contained in Qj as a single vertex component, and Pi = Gi /Si be obtained from P by deleting all edges of Qj . Obviously, for each {i, j} = {1, 2}, each pair of Gi and Si satisfies the description of the lemma, and Pi remains as a smallest bridgeless parity subgraph of Gi /Si . Thus, the cubic graph Gi is 3-edge-colorable since it is smaller than the smallest counterexample G. Properly renaming the colors of G1 (if necessary), a combination of 3-edge-colorings of G1 and G2 yields a 3edge-coloring of G.

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Circuit Double Cover of Graphs by Professor Cun-Quan Zhang

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