γ-labelings of complete bipartite graphs (original) (raw)

Near α-labelings of bipartite graphs

Australas. J Comb., 2000

An a-labeling of a bipartite graph G with n edges easily yields both a cyclic G-decomposition of Kn,n and of K 2nx +1 for all positive integers x. A ,B-Iabeling (or graceful labeling) of G yields a cyclic decomposition of K 2n + 1 only. It is well-known that certain classes of trees do not have a-Iabelings. In this article, we introduce the concept of a near a-labeling of a bipartite graph, and prove that if a graph G with n edges has a near a-labeling, then there is a cyclic G-decomposition of both Kn,n and K 2nx +1 for all positive integers x. We conjecture that all trees have a near a-labeling and show that certain classes of trees which are known not to have an a-labeling have a near a-labeling. 1 Introduction Only graphs without loops and without multiple edges will be considered herein. Undefined graph-theoretic terminology can be found in the textbook by Chartrand and Lesniak [1]. If m and n are integers with m ::; n we denote {m, m + 1, ... , n} by [m, n]. Let N denote the set of nonnegative integers and Zn the group of integers modulo n. If we consider Km to have the vertex set Zm, by clicking we mean applying the isomorphism i-+ i + 1. Likewise if we consider Km,m to have the vertex set Zm X Z2, with the obvious vertex bipartition, by clicking we mean applying the isomorphism (i,j)-+ (i + 1,j). Let K and G be graphs with G a subgraph of K. A G-decomposition of K is a set r = {G 1 , G 2 , ••• , G t } of subgraphs of K each of which is isomorphic to G and such that the edge sets of the graphs G i form a partition of the edge set of K. In this case, we say G divides K. If K is Km or Km,m, a G-decomposition r is cyclic (purely cyclic) if clicking is a permutation (t-cycle) of r. A labeling or valuation of a graph G is one-to-one function from V(G) into N. In 1967, Rosa [7] introduced several types of graph labelings as tools for decomposing

Totally Antimagic Total Labeling of Complete Bipartite Graphs

2017

For a graph G = (V,E) of order |V (G)| and size |E(G)| a bijection from the union of the vertex set and the edge set of G into the set {1, 2, . . . , |V (G)| + |E(G)|} is called a total labeling of G. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all edges incident with that vertex. The edge-weight of an edge is the sum of the label of the edge and the labels of the end vertices of that edge. A total labeling is called edge-antimagic (respectively, vertex-antimagic) if all edge-weights (respectively, vertex-weights) are pairwise distinct. If a total labeling is simultaneously edge-antimagic and vertex-antimagic at the same time, then it is called a totally antimagic total labeling. In this paper we prove that complete bipartite graphs admit totally antimagic total labeling. Mathematics Subject Classification (2010): 05C78

On totally antimagic total labeling of complete bipartite graphs

arXiv: Combinatorics, 2016

This paper deals with the problem of finding totally antimagic total labelings of complete bipartite graphs. We prove that complete bipartite graphs are totally antimagic total graphs. We also show that the join of complete bipartite graphs with one vertex is a totally antimagic total graph

Algorithms for Constructing Edge Magic Total Labeling of Complete Bipartite Graphs

International Journal of Computer and Communication Technology, 2015

The study of graph labeling has focused on finding classes of graphs which admits a particular type of labeling. In this paper we consider a particular class of graphs which demonstrates Edge Magic Total Labeling. The class we considered here is a complete bipartite graph Km,n. There are various graph labeling techniques that generalize the idea of a magic square has been proposed earlier. The definition of a magic labeling on a graph with v vertices and e edges is a one to one map taking the vertices and edges onto the integers 1,2,3,………, v+e with the property that the sum of the label on an edge and the labels of its endpoints is constant independent of the choice of edge. We use m x n matrix to construct edge magic total labeling of Km,n.

Compact Mean Labeling on Bipartite Graphs

International Journal of Scientific Research in Science and Technology, 2022

Labeling of graphs is the procedure of assigning numbers to the nodes, lines, or both in accordance with an applicable rule. In this study, we demonstrate that the bipartite graph K_(n,n) is a compact mean-labeled graph. We also explored graphs K_2,2 and K_(2,3 ) 〖,K〗_2,4 are compact mean-labeled graphs

Balanced Degree-Magic Labelings of Complete Bipartite Graphs under Binary Operations

Iranian Journal of Mathematical Sciences and Informatics, 2018

A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1, 2, ..., |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to (1 + |E(G)|) deg(v)/2. Degree-magic graphs extend supermagic regular graphs. In this paper we find the necessary and sufficient conditions for the existence of balanced degree-magic labelings of graphs obtained by taking the join, composition, Cartesian product, tensor product and strong product of complete bipartite graphs.

On some numerical characteristics of a bipartite graph

The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these characteristics of the set of all bipartite graphs of the type g=langleRgcupCg,Egrangleg=\langle R_g \cup C_g, E_g \rangleg=langleRgcupCg,Egrangle is formulated and proved, where V=RgcupCgV=R_g \cup C_gV=RgcupCg is the set of vertices, EgE_gEg is the set of edges of the graph ggg, $ |R_g |=m\ge 1$, ∣Cg∣=nge1|C_g |= n\ge 1∣Cg∣=nge1, ∣Eg∣=kge0|E_g |=k\ge 0∣Eg∣=kge0, m,nm,nm,n and kkk are integers.

On the Edge-balanced Index Sets of Complete Bipartite Graphs

2011

Let G be a graph with vertex set V (G) and edge set E(G), and f be a 0−1 labeling of E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling f edge-friendly. The edge-balanced index set of the graph G, EBI(G), is defined as the absolute difference between the number of vertices incident to more edges labeled 1 and the number of vertices incident to more edges labeled 0 over all edge-friendly labelings f. In 2009, Lee, Kong, and Wang [5] found the EBI(K l,n) for l = 1, 2, 3, 4, 5 as well as l = n. We continue the investigation of the EBI of complete bipartite graphs of other orders.