The genera of amalgamations of graphs

Abstract

If p ≤ m p \leq m , n then K m ∨ K p K n {K_m}{ \vee _{{K_p}}}{K_n} is the graph obtained by identify ing a copy of K p {K_p} contained in K m {K_m} with a copy of K p {K_p} contained in K n {K_n} . It is shown that for all integers p ≤ m p \leq m , n the genus g ( K m ∨ K p K n ) g({K_m}{ \vee _{{K_p}}}{K_n}) of K m ∨ K p K n {K_m}{ \vee _{{K_p}}}{K_n} is less than or equal to g ( K m ) + g ( K n ) g({K_m}) + g({K_n}) . Combining this fact with the lower bound obtained from the Euler formula, one sees that for 2 ≤ p ≤ 5 , g ( K m ∨ K p K n ) 2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n}) is either g ( K m ) + g ( K n ) g({K_m}) + g({K_n}) or else g ( K m ) + g ( K n ) − 1 g({K_m}) + g({K_n}) - 1 . Except in a few special cases, it is determined which of these values is actually attained.

The average Euler-genus of the vertex-amalgamation of signed graphs

“…, Σ k over U by Σ = U Σ i as the signed graph obtained by identifying distinct copies of U. For minimum genus and maximum genus of vertexamalgamation of graphs, see [1,2,3,12,13,29], etc.…”

The Genera of Edge Amalgamations of Complete Bigraphs

“…By(1), if there exists a 2-cell imbedding of G in a surface of genus g, then qiG; g) = 2, >.ik -4)F,. Hence if qiG; g) > 0, there is at least one 72-sided face P with »2 > 6.…”