Solution to some open problems on E-super vertex magic labeling of disconnected graphs

Abstract

An E-super vertex magic labeling is a bijection f :V(G)UE(G)→{1,2,3,…,p+q} such that for each vertex u, $f\left(u\right)+{\sum }_{v\in N\left(u\right)}f\left(uv\right)=k$ for some constant k where f(E(G))={1,2,3,…,q}. A graph that admits an E-super vertex magic labeling is called an E-super vertex magic graph. The only disconnected graphs that have been shown to be E-super vertex magic are mC_n if and only if both m and n are odd. The article “Marimuthu and Balakrishnan (2012)” discussed the E-super vertex magicness of connected graphs. In this paper, we pay our attention to prove the existence and non existence of E-super vertex magic labeling for some families of disconnected graphs. Also we provide solution to some open problems found in the article “Gray and MacDougall (2009)”.

Introduction

Throughout this paper, we restrict our attention to finite simple undirected graphs. The set of vertices and the set of edges of a graph G will be denoted by V (G) and E (G) respectively, and let p= |V(G)| and q=|E(G)|. The set of neighbors of a vertex v is denoted by N(v). For general graph theoretic notations, we follow [10].

A graph labeling is a mapping that carries a set of graph elements (usually vertices and/or edges) into a set of numbers (usually integers). Many kinds of labeling have been studied and an excellent survey of graph labeling can be found in [2].

Sedlàček [11] introduced the concept of magic labeling. Suppose that G is a graph with q edges. A graph G is magic if the edges of G can be labeled by the numbers 1, 2, 3,…,q so that the sums of all incident edge labels with every vertex are the same.

MacDougall et al. [7] introduced the notion of vertex magic total labeling. They studied the basic properties of vertex magic graphs and showed that some families of graphs have a vertex magic total labeling.

MacDougall et al. [8] introduced the notion of super vertex-magic total labeling. They call a vertex magic total labeling is super if f (V (G)) = {1, 2, 3,…, p}, we call it as V-super vertex magic labeling. In this labeling, the smallest labels are assigned to the vertices. They showed that a (p, q)-graph that has a super vertex-magic total labeling with magic constant k satisfies the following conditions: k = (p + q)(p+q+1)/p-(p+1)/2; k≥(41p+21)/8; if G is connected, k≥(7p−5)/2;p divides q(q+1) if p is odd, and p divides 2q(q+1) if p is even; if G has even order either p0 (mod 8) and q0 or 3 (mod 4) or p4(mod8) and q1 or 2 (mod 4); if G is r-regular and p and r have opposite parity, then p0 (mod 8) implies q0 (mod 4).They also show: C_n has a super vertex-magic total labeling if and only if n is odd; and no wheel, ladder, fan, friendship graph, complete bipartite graph or graph with a vertex of degree 1 has a super vertex-magic total labeling. They conjectured that no tree has a super vertex magic total labeling and that K_4n has a super vertex magic total labeling when n > 1. The later conjecture was proved by Gómez in [5].

Swaminathan and Jeyanthi [13] introduced a concept with same name of super vertex magic labeling. They call a vertex magic total labeling is super if f(E(G)) = {1, 2,…, q}. Note that the smallest labels are assigned to the edges. They proved the following: P_n is super vertex magic if and only if n is odd and n > 3; C_n is super vertex magic if and only if n is odd; the star graph is super vertex magic if and only if it is P₂; and mC_n is super vertex magic if and only if m and n are odd. In [12], they proved the following: no super vertex magic total graph has two or more isolated vertices or an isolated edge; a tree with n internal edges and tn leaves is not super vertex magic total if t >$\frac{n+1}{n}$; the graph obtained from a comb by attaching a pendant edge to each vertex of degree 2 is super vertex-magic total; the graph obtained by attaching a path with t edges to a vertex of an n-cycle is super vertex magic total if and only if n + t is odd. The use of the word “super” was introduced in [1].

MacDougall et al. [8] and Swaminathan and Jeyanthi [13] introduced different labelings with same name super vertex magic total labeling. To avoid confusion, Marimuthu and Balakrishnan [9] called a vertex magic total labeling E-super if f(E(G)) = {1, 2, 3,…,q}. Note that the smallest labels are assigned to the edges. A graph G is called E-super vertex magic if it admits an E-super vertex magic labeling.

The only disconnected graphs that have been shown to be E-super vertex magic are mC_n, where both m and n are odd [13]. In this paper, we investigate the properties of E-super vertex magic labeling, and exhibit some families of disconnected graphs that are E-super vertex magic and others that are not. Also we prove some open problems on E-super vertex magic labeling of disconnected graphs which is found in [4]. The following results that will subsequently be very useful to prove some theorems.

Definition 1.1

An n-sun is a cycle C_n with an edge terminating in a vertex of degree 1 attached to each vertex.

Lemma 1.2

([13]). If a non- trivial graph G is super vertex magic, then the magic constant k is given by k = q$+\frac{p+1}{2}$ +$\frac{q\left(q+1\right)}{p}$.

Theorem 1.3

([13]). mC_n is super vertex magic if and only if both m and n are odd.

Theorem 1.4

([6]). For each positive integer t ≥ 2, the disjoint union C₅U(2t)C₃ has a V-super vertex magic labeling with magic constant 21t+19.

Theorem 1.5

([6]). For each positive integer t ≥ 3, the disjoint union C₄U(2t-1)C₃ has a V-super vertex magic labeling with magic constant 21t+5.