By R. Balakrishnan, K. Ranganathan

Graph concept skilled a major progress within the twentieth century. one of many major purposes for this phenomenon is the applicability of graph conception in different disciplines equivalent to physics, chemistry, psychology, sociology, and theoretical computing device technology. This textbook presents a great heritage within the simple themes of graph conception, and is meant for a complicated undergraduate or starting graduate path in graph theory.

This moment variation contains new chapters: one on domination in graphs and the opposite at the spectral houses of graphs, the latter together with a dialogue on graph power. The bankruptcy on graph hues has been enlarged, protecting extra themes reminiscent of homomorphisms and colours and the distinctiveness of the Mycielskian as much as isomorphism. This booklet additionally introduces numerous fascinating subject matters similar to Dirac's theorem on k-connected graphs, Harary-Nashwilliam's theorem at the hamiltonicity of line graphs, Toida-McKee's characterization of Eulerian graphs, the Tutte matrix of a graph, Fournier's evidence of Kuratowski's theorem on planar graphs, the evidence of the nonhamiltonicity of the Tutte graph on forty six vertices, and a concrete program of triangulated graphs.

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**Example text**

Conversely, suppose that e D uv is a cut edge of G: Then the deletion of u results in the deletion of the edge uv: Since G is cubic, G u is disconnected. 2. Find the vertex cuts and edge cuts of the graph of Fig. 2. 3. G/ 3: Then G has a cut edge if and only if G has a cut vertex. 4. Show that in a graph, the number of edges common to a cycle and an edge cut is even. 3 Connectivity and Edge Connectivity We now introduce two parameters of a graph that in a way measure the connectedness of the graph.

1. G/ is partitioned into k nonempty subsets V1 ; V2 ; : : : ; Vk ; such that the induced subgraphs GŒV1 ; GŒV2 ; : : : ; GŒVk are all totally disconnected. It is said to be complete if, for i ¤ j; each vertex of Vi is adjacent to every vertex of Vj ; 1 Ä i; j Ä k: A k-partite tournament is an oriented complete k-partite graph (see Fig. 6). The subsets V1 ; V2 ; : : : ; Vk are often referred to as the partite sets of G: The next three theorems are based on Goddard et al. [74]. We now give a characterization of a k-partite tournament containing a 3-cycle.

In this chapter, we study the two graph parameters, namely, vertex connectivity and edge connectivity. We also introduce the parameter cyclical edge connectivity. We prove Menger’s theorem and several of its variations. In addition, the theorem of Ford and Fulkerson on flows in networks is established. 2 Vertex Cuts and Edges Cuts We now introduce the notions of vertex cuts, edge cuts, vertex connectivity, and edge connectivity. R. Balakrishnan and K. 1007/978-1-4614-4529-6 3, © Springer Science+Business Media New York 2012 49 50 Fig.