Magic graphs II
Abstract
Graph labelings is an active area of research in Graph Theory. There are many types of graph labelings which have been considered in recent years. A graph G(p, q) is said to be (1,1) edge-magic with the common edge count k0 if there exists a bijection f : V(G) ∪ E(G) →{1,... ,p + q} such that f(u) + f(v) + f(e) = k0 for all e = (u,v) ∈ E(G). A graph G(p,q) is said to be (1,1) vertex-magic with the common vertex number count k1 if there exists a bijection f : V(G) ∪ E(G) → {1,... ,p + q} such that for each u ∈ V(G), f(u) + ∑ e f(e) = k1 for all e = (u, v) ∈ E(G) with v ∈ V(G). A graph G(p, q) is said to be (1,0) edge-magic with the common edge count k2 if there exists a bijection f : V(G) -{1,... ,p} such that for all e = (u,v) ∈ E(G), f(u) + f(v) = k2. A graph G(p,q) is said to be (0,1) vertex-magic with the common vertex count k3 if there exists a bijection f : E(G) - {1,..., q} such that for each u ∈ V(G), ∑ e f(e) = k3 for all e = (u,v) ∈ E(G) with u ∈ V(G). A graph G(p,q) is said to be (1,0) vertex-magic with the common vertex count k4 there exists a bijection f : V(G) → {1,... ,p} such that for each u ∈ V(G), f(u) + f(v) = k4 for all v ∈ V(G) such that (u,v) ∈ E(G). A graph G(p,q) is said to be (0,1) edge-magic with the common edge-count k5 if there exists a bijection f : E(G) → {1,..., q} such that for each e ∈ E(G), f(e)+f(e0) = k5 for all e ∈ E(G) such that e and e0 are adjacent in G. The author has introduced a variety of graph labelings in [12]. In this paper, a number of interesting general results concerning these labelings are obtained.
