Francesc A. Muntaner-Batle

The place of super edge-magic labelings among other classes of labelings

Abstract A ( p , q )-graph G is edge-magic if there exists a bijective function f : V ( G )∪ E ( G )→{1,2,…, p + q } such that f ( u )+ f ( v )+ f ( uv )= k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f ( V ( G ))={1,2,…, p }. In this paper, we present some necessary conditions for a graph to be super edge-magic. By means of these, we study the super edge-magic properties of certain classes of graphs. We also exhibit the relationships between super edge-magic labelings and other well-studied classes of labelings. In particular, we prove that every super edge-magic ( p , q )-graph is harmonious and sequential (for a tree or q ⩾ p ) as well as it is cordial, and sometimes graceful. Finally, we provide a closed formula for the number of super edge-magic graphs.

A magical approach to some labeling conjectures

In this paper, a complete characterization of the (super) edge-magic linear forests with two components is provided. In the process of establishing this characterization, the super edgemagic, harmonious, sequential and felicitous properties of certain 2-regular graphs are investigated, and several results on super edge-magic and felicitous labelings of unions of cycles and paths are presented. These labelings resolve one conjecture on harmonious graphs as a corollary, and make headway towards the resolution of others. They also provide the basis for some new conjectures (and a weaker form of an old one) on labelings of 2-regular graphs.

Labeling the vertex amalgamation of graphs

A graph G of size q is graceful if there exists an injective function f : V (G) → {0, 1, . . . , q} such that each edge uv of G is labeled |f(u)− f(v)| and the resulting edge labels are distinct. Also, a (p, q) graph G with q ≥ p is harmonious if there exists an injective function f : V (G) → Zq such that each edge uv of G is labeled f(u)+f(v) (mod q) and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function f : V (G) → Zq+1 such that each edge uv of G is labeled f(u) + f(v) (mod q) and the resulting edge labels are distinct. In this paper, we present several results involving the vertex amalgamation of graceful, felicitous and harmonious graphs. 130 R.M. Figueroa-Centeno, R. Ichishima, F.A. Muntaner-Batle Further, we partially solve an open problem of Lee et al., that is, for which m and n the vertex amalgamation of n copies of the cycle Cm at a fixed vertex v ∈ V (Cm), Amal(Cm, v, n), is felicitous? Moreover, we provide some progress towards solving the conjecture of Koh et al., which states that the graph Amal(Cm, v, n) is graceful if and only if mn ≡ 0 or 3 (mod 4). Finally, we propose two conjectures.

On the Super Edge-Magic Deficiency of Graphs

Abstract A (p, q) graph G is edge-magic if there exists a bijective function f : V ( G ) ∪ E ( G ) → {1,2,…, p + q } such that f ( u ) + f ( v ) + f ( uv ) = k is a constant, called the valence of f , for any edge uv of G . Moreover, G is said to be super edge-magic if f ( V ( G )) = {1,2,…, p }. The question studied in this paper is for which graphs is it possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic? If it is possible for a given graph G , then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, μ s (G) of G ; otherwise we define it to be + ∞.

On the structure of paths-like trees

We study the structure of path-like trees. In order to do this, we introduce a set of trees that we call expandable trees. In this paper we also generalize the concept of path-like trees and we call such generalization generalized path-like trees. As in the case of path-like trees, generalized path-like trees, have very nice labeling properties.