Bounds for the average genus of the vertex-amalgamation of graphs

Abstract The average orientable genus of graphs has been the subject of a considerable number of recent investigations. It is the purpose of this article to examine the extent to which the average genus of the amalgamation of graphs fails to be additive over its constituent subgraphs. This discrepancy is bounded and the sharpness of these bounds discussed and compared to similar bounds for the minimum and the maximum genera of the amalgamation of graphs. The main result relies on permutation-partition pairs for its proof.

[1]  Henry H. Glover,et al.  The genus of the 2-amalgamations of graphs , 1981, J. Graph Theory.

[2]  Jianer Chen,et al.  Lower bounds for the average genus , 1995, J. Graph Theory.

[3]  Richard D. Ringeisen,et al.  A Kuratowski-type theorem for the maximum genus of a graph , 1972 .

[4]  S. R. Alpert.  The genera of amalgamations of graphs , 1973 .

[5]  Saul Stahl.  Permutation-partition pairs: a combinatorial generalization of graph embeddings , 1980 .

[6]  Jianer Chen,et al.  Limit points for average genus. I. 3-Connected and 2-connected simplicial graphs , 1992, J. Comb. Theory, Ser. B.

[7]  Saul Stahl,et al.  Region distributions of some small diameter graphs , 1991, Discret. Math..

[8]  Jianer Chen,et al.  Limit points for average genus II. 2-Connected non-simplicial graphs , 1992, J. Comb. Theory, Ser. B.

[9]  A. White.  Graphs, Groups and Surfaces , 1973 .

[10]  Charles H. C. Little,et al.  An additivity theorem for maximum genus of a graph , 1978, Discret. Math..

[11]  Saul Stahl,et al.  Permutation-partition pairs. III. Embedding distributions of linear families of graphs , 1991, J. Comb. Theory, Ser. B.

[12]  Jonathan L. Gross,et al.  Hierarchy for imbedding-distribution invariants of a graph , 1987, J. Graph Theory.

[13]  Saul Stahl.  Permutation-partition pairs. II. Bounds on the genus of the amalgamation of graphs , 1982 .

[14]  F. Harary,et al.  Additivity of the genus of a graph , 1962 .

[15]  G. Ringel.  Map Color Theorem , 1974 .

[16]  Saul Stahl.  On the average genus of the random graph , 1995, J. Graph Theory.

[17]  Jianer Chen,et al.  Kuratowski-Type Theorems for Average Genus , 1993, J. Comb. Theory, Ser. B.

[18]  Saul Stahl,et al.  Region distributions of graph embeddings and stirling numbers , 1990, Discret. Math..

[19]  Saul Stahl.  An upper bound for the average number of regions , 1991, J. Comb. Theory, Ser. B.

[20]  Dan Archdeacon,et al.  The orientable genus is nonadditive , 1986, J. Graph Theory.

[21]  J. Kiefer,et al.  On the theory of queues with many servers , 1955 .