By Gary Chartrand, Ping Zhang
Written through of the main well-liked figures in the sector of graph thought, this entire text provides a remarkably student-friendly procedure. aimed toward undergraduates taking a primary direction in graph idea, its sound but obtainable remedy emphasizes the heritage of graph thought and offers unique examples and lucid proofs. 2004 version.
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Extra info for A First Course in Graph Theory (Dover Books on Mathematics)
Pseudosurfaces We now consider topological spaces akin to surfaces, but which fail to be 2-manifolds at a finite number of points; these spaces form additional candidates for the imbedding of graphs, and were studied extensively by Petroelje [P5]. Def. 5-28. Let A denote a set of ]C*=1 nirni ^ 0 distinct points of Sk-, with 1 < mi < 7TT-2 < • • • < rat. Partition A into HI sets of ra^ points each, i — 1,2, • • • , t. For each set of the partition, identify all the points of that set. S(fc;rai (mi), 712(7712), • • • ,nt(mt}).
A 3-connected planar graph is uniquely imbeddable on the sphere. One readily verifies that the five planar graphs of Figure 5-8 are also 3-connected. The following theorem of Weinberg [W3] gives information about the automorphism groups of 3-polytopal graphs: 5-4. THREE APPLICATIONS 45 Thm. 5-26. Let G be 3-polytopal, with q edges. Then | Aut(G)| < 4g, with equality holding if and only if G is the 1-skeleton of a Platonic Solid. The Greeks identified four of the Platonic Solids with the four basic elements: earth with hexahedron, air with octahedron, fire with tetrahedron (the sharpest of the solids), and water with icosahedron (the roundest); the dodecahedron became the all-encompassing universe.
If so, what is its index? ) For k > 2, the odd graph Ok has all (k — l)-subsets of a (2k — 1)set as vertices, with adjacency corresponding to the property of being disjoint. Then 02 = K3 and Oa = II, the Petersen graph. ) It is immediate that S^k-i < Aut(Ofc); show that in fact Sat-i = Aut(Ofc). In particular, Aut(II) = S5. This page intentionally left blank CHAPTER 4 THE CAYLEY COLOR GRAPH OF A GROUP PRESENTATION In this chapter we see that each group can be defined in terms of generators and relations and that corresponding to such a presentation there is a unique graph, called the Cayley color graph of the presentation.
A First Course in Graph Theory (Dover Books on Mathematics) by Gary Chartrand, Ping Zhang