Alpha Labeling of Amalgamated Cycles

A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of Eulerian graphs, and edge amalgamation to generate a family of outerplanar graphs. In addition, we determine the number of graphs obtained with $k$ copies of the cycle $C_n$, for both types of amalgamations

The game L(d,1)-labeling problem of graphs

AbstractLet G be a graph and let k be a positive integer. Consider the following two-person game which is played on G: Alice and Bob alternate turns. A move consists of selecting an unlabeled vertex v of G and assigning it a number a from {0,1,2,…,k} satisfying the condition that, for all u∈V(G),u is labeled by the number b previously, if d(u,v)=1, then |a−b|≥d, and if d(u,v)=2, then |a−b|≥1. Alice wins if all the vertices of G are successfully labeled. Bob wins if an impasse is reached before all vertices in the graph are labeled. The game L(d,1)-labeling number of a graph G is the least k for which Alice has a winning strategy. We use λ̃1d(G) to denote the game L(d,1)-labeling number of G in the game Alice plays first, and use λ̃2d(G) to denote the game L(d,1)-labeling number of G in the game Bob plays first. In this paper, we study the game L(d,1)-labeling numbers of graphs. We give formulas for λ̃1d(Kn) and λ̃2d(Kn), and give formulas for λ̃1d(Km,n) for those d with d≥max{m,n}