Download PDF by Lars Dovling Andersen, Ivan Tafteberg Jakobsen, Carsten: Graph Theory in Memory of G.A. Dirac

By Lars Dovling Andersen, Ivan Tafteberg Jakobsen, Carsten Thomassen, Bjarne Toft and Preben Dahl Vestergaard (Eds.)

ISBN-10: 0444871292

ISBN-13: 9780444871299

This quantity is a tribute to the existence and mathematical paintings of G.A. Dirac (1925-1984). one of many top graph theorists, he built tools of significant originality and made many basic discoveries. The forty-two papers are all fascinated with (or on the topic of) Dirac's major strains of analysis. a few mathematicians pay tribute to his reminiscence through providing new leads to various components of graph concept. one of the themes incorporated are paths and cycles, hamiltonian graphs, vertex colouring and important graphs, graphs and surfaces, edge-colouring, and endless graphs. a number of the papers have been initially offered at a gathering held in Denmark in 1985. Attendance being via invitation basically, a few fifty five mathematicians from 14 international locations participated in quite a few lectures and discussions on graph concept concerning the paintings of Dirac. This quantity comprises contributions from others besides, so shouldn't be seemed in simple terms because the court cases of that assembly. A difficulties part is integrated, in addition to a list of Dirac's personal courses.

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The last part of (11) follows by directional duality. 2). (12) Proof of (12): Suppose A1 # {yz}. We must have that A1 \ {y2} dominates a (and thus a -+ y2), since otherwise (4) and (11) imply, that T has an (zz,ya)-path, which is disjoint from the path 21 -+ a w -+ y -+ y1, where a -+ w and w E A l , contradicting (*). (Note that by (6), A1 dominates y). Now A1 \ {yz} dominates a, 21,y. Thus by the 3-connectivity of T, there exists an edge 2 -+ u, where u E A1 \ {yz}. , 21 -+ x --+ u -+ b -+ y1 and 2 2 -+ y -+ a y2 are disjoint paths by (8) and (ll),contradicting (*).

This means: the minimal degree of G is at least 3. The first property of G follows directly from the definition of M(3,>1). The second property of G follows from the >I-minimality of G. The last conclusion is not trivial, for its validity is based on the property of 3, which is called preservation for embedding of 3. The concept of preservation of embedding of F is defined on the following way: Let G and H be two arbitrary graphs with G > 1 H . A surface F is said to preserve embedding with respect to >1 if, and only if, the embeddability of G in 3 implies the embeddability of H in F.

D. Parsons, “A construction for vertex-transitive graphs,” Canad. J. Math. 34 (1982), 307-318. [3] B. Alspach, E. Durnberger and T. D. Parsons, “Hamilton cycles in metacirculant graphs with prime cardinality blocks,” Annals of Discrete Math. 27 (1985), 27-34. [4] K. Bannai, “Hamiltonian cycles in generalized Petersen graphs,” J. Combin. Theory ( B ) 24 (1978), 181-188. [5] C. C. Chen and N. F. Quimpo, “On strongly hamiltonian abelian group graphs,” Combinatorial Mathematics VIII (K. ), Lecture Notes in Math.

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Graph Theory in Memory of G.A. Dirac by Lars Dovling Andersen, Ivan Tafteberg Jakobsen, Carsten Thomassen, Bjarne Toft and Preben Dahl Vestergaard (Eds.)

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