MT4514: Graph Theory

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40 Colva M. Roney-DougalThe graphs G 1 and G 2 have fewer edges than G, because there is at least one edgein U that is not in S, and at least one edge in W that is not in S. Since S is aminimal Aw-disconnecting set in G 1 and vZ-disconnecting set in G 2 , the inductionhypothesis tells us that there are k edge-disjoint paths in G 1 from A to w and inG 2 from v to Z. The required k edge-disjoint paths in G are then obtained bycombining these paths in the obvious way.Case 2: Suppose that every vw-disconnecting set S of size k consists only of edgesthat are all incident to v or all incident to w. We can assume by induction thatevery edge of G is contained in one of these minimum vw-disconnecting sets, sinceotherwise its removal would not affect the value of k and we could use the inductionhypothesis to obtain k edge-disjoint paths. It follows that if P is any path fromv to w then P must consist of either one or two edges, and therefore contains atmost one edge of any vw-disconnecting set of size k. By removing all edges in Pfrom G, we obtain a graph which contains at least k − 1 edge-disjoint paths (by theinduction hypothesis). These (k − 1) paths, together with P , give the required kpaths in G. Theorem 4.5 (Menger’s theorem (vertex version)) Let v and w be distinctnonadjacent vertices of a graph G. The maximum number of vertex disjoint pathsfrom v to w is equal to the minimum number of vertices in a vw-separating set.Exercise 4.6. The proof of the vertex version of Menger’s theorem is very similarto that of the edge version, try to write it down.We finish by showing that Hall’s Marriage Theorem 1.4 can be deduced fromthe vertex version of Menger’s theorem.Theorem 4.7. Theorem 4.5 implies Hall’s Marriage Theorem 1.4.Proof. Let G be a connected bipartite graph with vertex sets V 1 and V 2 . Wehave to show that it follows from Theorem 4.5 that if each set A ⊂ V 1 of verticeshas at least |A| neighbours then there exists a matching from V 1 to V 2 . Make a newgraph G 1 by adjoining to G a vertex v that is adjacent to every vertex in V 1 and avertex w that is adjacent to every vertex in V 2 .
<strong>Graph</strong> <strong>Theory</strong> 41A matching from V 1 to V 2 exists if and only if the number of vertex-disjoint pathsfrom v to w is equal to the number of vertices in V 1 . Let |V 1 | = k. By Theorem 4.5,it is therefore enough to show that every vw-separating set contains at least kvertices.Let S be a vw-separating set, consisting of a subset A of V 1 and a subset Bof V 2 . Since S = A ∪ B is a vw-separating set, there are no edges joining verticesin V 1 \ A to vertices in V 2 \ B, otherwise there would be paths using these edgesbetween v and w. Therefore, the neighbours of V 1 \ A that lie in V 2 must all be inB. By assumption, the set V 1 \ A of vertices has at least |V 1 \ A| neighbours in V 2 .We deduce that|V 1 \ A| ≤ |B|,so |S| = |A| + |B| ≥ |A| + |V 1 \ A| = |V 1 | = k, as required.
Page 2 and 3: Contents1 Introduction 31 About the
Page 4 and 5: 4 Colva M. Roney-DougalEuler in 173
Page 6 and 7: 6 Colva M. Roney-DougalNow consider
Page 8 and 9: Chapter 2Basic definitions and isom
Page 10: 10 Colva M. Roney-DougalDefinition
Page 13 and 14: Graph Theory 13Then none of the ver
Page 15 and 16: Chapter 3Paths on graphs1. Definiti
Page 17 and 18: 173. Здравствена уст
Page 20 and 21: 20 Colva M. Roney-DougalLet R be th
Page 22: 22 Colva M. Roney-DougalThe value o
Page 25 and 26: Graph Theory 25The theorem was firs
Page 27 and 28: Graph Theory 27and ⌊x⌋ is the l
Page 29 and 30: Graph Theory 29Remark 5.6. The chro
Page 31 and 32: Graph Theory 31Proof. Break the pol
Page 33 and 34: Chapter 5Marriage, Matchings and Co
Page 35 and 36: Graph Theory 35k-edge-colourable if
Page 37 and 38: Graph Theory 37We check that no edg
Page 39: Graph Theory 39In general we will b
Page 43 and 44: Graph Theory 43(3) ⇒ (4): The rem
Page 45 and 46: Graph Theory 451. The vertices of d
Page 47 and 48: Graph Theory 477. Our new list is [
Page 49 and 50: Graph Theory 49This completes the i
Page 51 and 52: Graph Theory 51Challenges Find R(3,
Page 53 and 54: Chapter 8Adjacency matricesAim: To
Page 55 and 56: Graph Theory 55Proof. Denote the ei
Page 57 and 58: Graph Theory 57since v = 1 + d 2 .B
Page 59: Graph Theory 59sothat is(t 2 + 7)(t

MT4514: Graph Theory

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