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 symmetric group action on the maximal chains in a finite, ranked poset is local if the adjacent transpositions act in such a way that (i, i+ 1) sends each maximal chain either to itself or to one differing only at rank i. We prove that when Sn acts locally on a lattice, each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by label...

A class TC(X) of X-graphs is introduced, and an algorithmic property of labellings of nite strongly connected graphs by rational languages is shown to be decidable. This property, named as TC-inevitability, is a language-theoretic analogue of the R-inevitability of labellings of nite strongly connected graphs by nite monoids. As a consequence, the pseudovariety R of all nite R-trivial monoids i...

Is it possible to label the edges of Kn with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question’s perhaps surprising affirmative answer. More generally, we address the question that arises when “Hamilton cycle” is replaced by “k-factor” for nonnegative integers k. Such edge-labelling...

In this paper we define total magic cordial (TMC) and total sequential cordial (TSC) labellings which are weaker versions of magic and simply sequential labellings of graphs. Based on these definitions we have given several results on TMC and TSC graphs.

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...

where E(v) is the set of edges that have v as an end-point. The total labelling λ of G is vertex-magic if every vertex has the same weight, and the graph G is vertexmagic if a vertex-magic total labelling of G exists. Magic labellings of graphs were introduced by Sedlácěk [5] in 1963, and vertex-magic total labellings first appeared in 2002 in [4]. For a dynamic survey of various forms of graph...

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...