The Genera of Amalgamations of Graphs
Seth R. Alpert · Transactions of the American Mathematical Society · 1973
If $p \leq m$, n then ${K_m}{ \vee _{{K_p}}}{K_n}$ is the graph obtained by identify ing a copy of ${K_p}$ contained in ${K_m}$ with a copy of ${K_p}$ contained in ${K_n}$ . It is shown that for all integers $p \leq m$, n the genus $g({K_m}{ \vee _{{K_p}}}{K_n})$ of ${K_m}{ \vee _{{K_p}}}{K_n}$ is less than or equal to $g({K_m}) + g({K_n})$. Combining this fact with the lower bound obtained from the Euler formula, one sees that for $2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})$ is either $g({K_m}) + g({K_n})$ or else $g({K_m}) + g({K_n}) - 1$. Except in a few special cases, it is determined which of these values is actually attained.