An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Largest cliques in connected supermagic graphs

A graph $G=(V,E)$ is said to be $\textit{magic}$ if there exists an integer labeling $f: V \cup E \to [1, |V \cup E|]$ such that $f(x)+f(y)+f(xy)$ is constant for all edges $xy \in E$. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most $3n^2+o(n^2)$ which contain a complete graph of order $n$. Bounds on Sidon sets show that the order of such a graph is at least $n^2+o(n^2)$. We close the gap between those two bounds by showing that, for any given graph $H$ of order $n$, there are connected magic graphs of order $n^2+o(n^2)$ containing $H$ as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling $f$, which satisfies the additional condition $f(V)=[1,|V|]$

Largest cliques in connected supermagic graphs

A graph G = (V, E) is said to be magic if there exists an integer labeling f: V ∪ E − → [1, |V ∪ E|] such that f(x) + f(y) + f(xy) is constant for all edges xy ∈ E. Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n 2 + o(n 2) which contain a complete graph of order n. Bounds on Sidon sets show that the order of such a graph is at least n 2 + o(n 2). We close the gap between those two bounds by showing that, for any given graph H of order n, there are connected magic graphs of order n 2 + o(n 2) containing H as an induced subgraph. Moreover it can be required that the graph admits a supermagic labelling f, which satisfies the additional condition f(V) = [1, |V |]