By R. Balakrishnan, K. Ranganathan
This moment version comprises new chapters: one on domination in graphs and the opposite at the spectral houses of graphs, the latter including a dialogue on graph energy. The bankruptcy on graph colours has been enlarged, protecting extra subject matters equivalent to homomorphisms and hues and the individuality of the Mycielskian as much as isomorphism.
This ebook 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 facts 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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Extra resources for A Textbook of Graph Theory
5. A vertex of G is a cut vertex of G if, and only if, it belongs to at least two blocks of G. 7) implies that a graph with at least three vertices is a block if, and only if, any two vertices of the graph are connected by at least two internally disjoint paths. 8, we see that any two vertices of a block with at least three vertices belong to a common cycle. Thus, a block with at least three vertices contains a cycle.
25 are forbidden subgraphs for G. Howe ver, for the sake of later reference, we pro ve the follo wing result. 6 lfG is a line graph, then K I,3 is aforbidden subgraph of G. I. 25. Ninegraphs of Bieneke  Proof Suppose that G is the line graph of graph H, and that G contains K 1,3 as an induced subgraph. If v is the vertex of degree 3 in K 1,3 and vI , V2, and V3 are the neighbors of v of this K1,3 , then the edge e corresponding to v in H is adjacent to the three edges el , e 2, and e3 corresponding to the vertices VI , V2, and V3.
Let T be a tournament on (n + 1) vertices VI , V2 , . , Vn+l ' Now delete Vn+ 1 from T . The resulting digraph T' is a tournament on n vertices and hence by the induction hypothesis contains a directed Hamilton path. Assume that the Hamilton path is VI V2 . . Vn , relabelin g the vertices, if necessary. If the arc joining VI and Vn+ l has vn +1 as its tail, then Vn+1 V I V2 . . Vn is a directed Hamilton path in T , and the result stands proved. ) If the arc joining Vn and Vn +1 is directed from Vn to Vn+l, then VI V2 .
A Textbook of Graph Theory by R. Balakrishnan, K. Ranganathan