Rikio Ichishima

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.

On Partitional Labelings of Graphs

The notion of partitional graphs, a subclass of sequential graphs, is introduced, and the cartesian product of a partitional graph and K2 is shown to be partitional. Every sequential graph is harmonious and felicitous. The partitional property of some bipartite graphs including the n-dimensional cube Qn is studied, and thus this paper extends what was known about the sequentialness, harmoniousness and felicitousness of such 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 beta-number of forests with isomorphic components

Abstract The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1, . . . , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c + 1, . . . , c + |E (G)| − 1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by βs (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m|V (G)|−1 if G has the additional property that G is a graceful nontrivial tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars.

New parameters for studying graceful properties of graphs

Abstract In this paper, we introduce two kinds of new parameters for studying graceful properties of graphs. We also establish the relation among these parameters. Moreover, we provide the relation between super edge-magic labelings and graceful labelings of trees.

Gracefully Cultivating Trees on a Cycle

Abstract A graph G of size q is graceful if there exists an injective function f : V ( G ) → { 0 , 1 , … , q } such that each u v ∈ E ( G ) is labeled | f ( u ) − f ( v ) | and the resulting edge labels are distinct. Truszczynski conjectured that all unicyclic graphs except the cycle C n , where n ≡ 1 or 2 ( mod 4 ) , are graceful. In this paper, we present two methods to construct certain graceful unicyclic graphs when the length of cycles are congruent to 0 or 3 (mod 4).