Supermagic Generalized Double Graphs 1

A graph G is called supermagic if it admits a labelling of the edges by pairwise di erent consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we will introduce some constructions of supermagic labellings of some graphs generalizing double graphs. Inter alia we show that the double graphs of regular Hamiltonian graphs and some circulant graphs are supermagic

Some constructions of supermagic graphs using antimagic graphs

Abstract. AgraphG is called supermagic if it admits a labelling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex, the weight of vertex, is independent of the particular vertex. A graph G is called (a, 1)-antimagic if it admits a labelling of the edges by the integers {1,...,|E(G)|} such that the set of weights of the vertices consists of different consecutive integers. In this paper we will deal with the (a, 1)-antimagic graphs and their connection to the supermagic graphs. We will introduce three constructions of supermagic graphs using some (a, 1)-antimagic graphs

M_{2}-edge colorings of dense graphs

An edge coloring $\varphi$ of a graph $G$ is called an $\mathrm{M}_i$-edge coloring if $|\varphi(v)|\leq i$ for every vertex $v$ of $G$, where $\varphi(v)$ is the set of colors of edges incident with $v$. Let $\mathcal{K}_i(G)$ denote the maximum number of colors used in an $\mathrm{M}_i$-edge coloring of $G$. In this paper we establish some bounds of $\mathcal{K}_2(G)$, present some graphs achieving the bounds and determine exact values of $\mathcal{K}_2(G)$ for dense graphs

Note on independent sets of a graph

summary:Let the number of $k$-element sets of independent vertices and edges of a graph $G$ be denoted by $n(G,k)$ and $m(G,k)$, respectively. It is shown that the graphs whose every component is a circuit are the only graphs for which the equality $n(G,k)=m(G,k)$ is satisfied for all values of $k$