Abstract
For every $h\in \mathbb{N}$, a graph $G$ with the vertex set $V(G)$ and the edge set $E(G)$ is said to be $h$-magic if there exists a labeling $l : E(G) \rightarrow\mathbb{Z}_h \setminus \{0\}$ such that the induced vertex labeling $s : V (G) \rightarrow \mathbb{Z}_h$, defined by $s(v) =\sum_{uv \in E(G)} l(uv)$ is a constant map. When this constant is zero, we say that $G$ admits a zero-sum $h$-magic labeling. The null set of a graph $G$, denoted by $N(G)$, is the set of all natural numbers $h \in \mathbb{ N} $ such that $G$ admits a zero-sum $h$-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if $G$ is an $r$-regular graph, then for even $r$ ($r > 2$), $N(G)=\mathbb{N}$ and for odd $r$ ($r\neq5$), $\mathbb{N} \setminus \{2,4\}\subseteq N(G)$. Moreover, we prove that if $r$ is odd and $G$ is a $2$-edge connected $r$-regular graph ($r\neq 5$), then $ N(G)=\mathbb{N} \setminus \{2\}$. Also, we show that if $G$ is a $2$-edge connected bipartite graph, then $\mathbb{N} \setminus \{2,3,4,5\}\subseteq N(G)$.
Highlights
Let G be a finite and undirected graph with vertex set V (G) and edge set E(G)
In this paper we show that if G is an r-regular graph, for even r (r > 2), N (G) = N and for odd r (r = 5), N \ {2, 4} ⊆ N (G)
Choi, Georges and Mauro [6] proved that if G is 3-regular graph, N (G) is N \ {2} or N \ {2, 4}. We extend this result by showing that if G is an rregular graph, for even r (r > 2), N (G) = N and for odd r (r = 5), N\{2, 4} ⊆ N (G)
Summary
Let G be a finite and undirected graph with vertex set V (G) and edge set E(G). A graph in which multiple edges are admissible is called a multigraph. A graph G is said to be zero-sum h-magic if there is an edge labeling from E(G) into Zh \ {0} such that the sum of values of all edges incident with each vertex is zero. The original concept of A-magic graph is due to Sedlacek [14], who defined it to be a graph with a real-valued edge labeling such that have distinct non-negative labels, and, in the manner described above, the sum of the labels of the edges incident to vertex v is constant over V (G). Theorem 5.[4, p.179] Let r 3 be an odd integer, and G be a 2-edge connected [r − 1, r]-. Theorem 6.[5] Every 2-edge connected (2r + 1)-regular multigraph contains a 2-factor.
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