On the super edge-magic deficiency of join product and chain graphs

A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to {1, 2, 3, ..., p+q} such that f(x) + f(xy) + f(y) is a constant for every edge $xy \in E(G)$ and f(V(G)) = {1, 2, 3, ..., p}. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or +~ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.

On D-distance (anti)magic labelings of shadow graph of some graphs

A graph G is called super edge-magic if there exists a bijection <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$f:V(G)\cup E(G)⟶\{1,2,\cdots ,|V(G)|+|E(G)|\}$, where <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$f(V(G))=\{1,2,\cdots ,|V(G)|\}$, such that <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$f(u)+f(uv)+f(v)$ is a constant for every edge <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$uv\in E(G)$. Such a case, f is called a super edge magic labeling of G. A bipartite graph G with partite sets A and B is called consecutively super edge-magic if there exists a super edge-magic labeling f with the property that <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$f(A)=\{1,2,\cdots ,|A|\}$ and <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$f(B)=\{|A|+1,|A|+2,\cdots ,|V(G)|\}$. The super edge-magic deficiency of a graph G, denoted by <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->${\mu }_s(G)$, is either the minimum nonnegative integer n such that <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$G\cup n{K}_1$ is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a bipartite graph G, denoted by <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->${\mu }_c(G)$, is either the minimum nonnegative integer n such that <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$G\cup n{K}_1$ is consecutively super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$2C_3\cup C_n$ and join product of <!--${MathJax: TeX-AMS-MML_HTMLorMML}-->$K_{1,n}\cup P_m$ with an isolated vertex.

Let G be a finite, simple, and undirected graph with vertex set V G and edge set E G . A super edge-magic labeling of G is a bijection f : V G ∪ E G ⟶ 1,2 , … , V G + E G such that f V G = 1,2 , … , V G and f u + f u v + f v is a constant for every edge u v ∈ E G . The super edge-magic labeling f of G is called consecutively super edge-magic if G is a bipartite graph with partite sets A and B such that f A = 1,2 , … , A and f B = A + 1 , A + 2 , … , V G . A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency of G , denoted by μ s G , is either the minimum nonnegative integer n such that G ∪ n K 1 is super edge-magic or + ∞ if there exists no such n . The consecutively super edge-magic deficiency of a graph G is defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.

Let G be a graph with n vertices and f : V(G)→{1, 2, 3, …, n} be a bijection. The {0, 1}-vertex weight of a vertex v ∈ V(G) under f is wf[v] = f(v) + Σ_u∈N(v) f(u). If wf[v] = k for each v ∈ V(G) then f is called a {0,1}-distance magic labeling of G with magic constant k. If {wf[v] : v ∈ V(G)} is {a, a + d, a + 2d, …, a + (n – 1)d} then f is called an (a, d)-{0,1}-distance antimagic labeling of G. A graph is called a {0, 1}-distance magic graph (resp. an (a, d)-{0,1}- distance antimagic graph), if it admits a {0, 1}-distance magic labeling (resp. an (a, d)-{0,1}-distance antimagic labeling). In this article, we completely solve the problem of {0, 1}-distance magic 2-regular graphs and show that several classes of these graphs are (a, d)-{0, 1}-distance antimagic.