Ryan Jones, Western Michigan University We introduce a modular edge-graceful labeling of a graph a dual concept to the common graceful labeling. A 1991 conjecture known as the Modular Edge-Graceful Tree Conjecture states that every tree of order n where n 6≡ 2 (mod 4) is modular edge-graceful. We show that this conjecture is true. More general results and questions on this topic are presented.

a graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ e(g)} consists of n integers. inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of g and k2 , and the disjoint :union: of g...

A graph of size n is said to be graceful when is possible toassign distinct integers from {0, 1, . . . , n} to its verticesand {|f(u)−f(v)| : uv ∈ E(G)} consists of n integers. Inthis paper we present broader families of graceful graphs; these families are obtained via three different operations: the third power of a caterpillar, the symmetric product of G and K2 , and the disjoint union of G a...

A graceful labeling of a graphG = (V,E) assigns |V | distinct integers from the set {0, . . . , |E|} to the vertices of G so that the absolute values of their differences on the |E| edges of G constitute the set {1, . . . , |E|}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful. We pro...

graceful labeling of G. Such sets P and Q are called the vertex values set and edge labels set of G, respectively. The notions of super-edge-graceful graphs and edge-graceful graphs are different. Mitchem and Simoson [6] showed that the step graph C2 6 is super-edge-graceful but not edge-graceful. Shiu [7] showed that the complete graph K4 is edge-graceful but not super-edge-graceful. Mitchem a...

Abstract In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.

This paper introduces the notion of discrete t-set graceful graphs and obtains some of their properties. It also examines the interrelations among different types of set-indexers, namely, set-graceful, set-semigraceful, topologically set-graceful (t-set graceful), strongly t-set graceful and discrete t-set graceful and establishes how all these notions are interdependent or not. AMS Mathematics...

Let [n]∗ denote the set of integers {−n−1 2 , . . . , n−1 2 } if n is odd, and {−n 2 , . . . , n 2 } \ {0} if n is even. A super edge-graceful labeling f of a graph G of order p and size q is a bijection f : E(G) → [q]∗, such that the induced vertex labeling f ∗ given by f ∗(u) = ∑ uv E(G) f(uv) is a bijection f ∗ : V (G) → [p]∗. A graph is super edge-graceful if it has a super edge-graceful la...