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 $\mathbb{N} \setminus \{2,3,4,5\}\subseteq N(G)$.
The vehicle routing problem (VRP) is a well-known NP-Hard problem in operation research which has drawn enormous interest from many researchers during the last decades because of its vital role in planning of distribution systems and logistics. This article presents a modified version of the elite ant system (EAS) algorithm called HEAS for solving the VRP. The new version mixed with insert and swap algorithms utilizes an effective criterion for escaping from the local optimum points. In contrast to the classical EAS, the proposed algorithm uses only a global updating which will increase pheromone on the edges of the best (i.e. the shortest) route and will at the same time decrease the amount of pheromone on the edges of the worst (i.e. the longest) route. The proposed algorithm was tested using fourteen instances available from the literature and their results were compared with other well-known meta-heuristic algorithms. Results show that the suggested approach is quite effective as it provides solutions which are competitive with the best known algorithms in the literature.
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