The Restricted Edge-Connectivity of Strong Product Graphs
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2024
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| _version_ | 1866911766275948544 |
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| author | Ye, Hazhe Tian, Yingzhi |
| author_facet | Ye, Hazhe Tian, Yingzhi |
| contents | The restricted edge-connectivity of a connected graph $G$, denoted by $λ^{\prime}(G)$, if it exists, is the minimum cardinality of a set of edges whose deletion makes $G$ disconnected and each component with at least 2 vertices. It was proved that if $G$ is not a star and $|V(G)|\geq4$, then $λ^{\prime}(G)$ exists and $λ^{\prime}(G)\leqξ(G)$, where $ξ(G)$ is the minimum edge-degree of $G$. Thus a graph $G$ is called maximally restricted edge-connected if $λ^{\prime}(G)=ξ(G)$; and a graph $G$ is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of $G$. The strong product of graphs $G$ and $H$, denoted by $G\boxtimes H$, is the graph with vertex set $V(G)\times V(H)$ and edge set $\{(x_1,y_1)(x_2,y_2)\ |\ x_1=x_2$ and $y_1y_2\in E(H)$; or $y_1=y_2$ and $x_1x_2\in E(G)$; or $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$\}. In this paper, we determine, for any nontrivial connected graph $G$, the restricted edge-connectivity of $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$, where $P_n$, $C_n$ and $K_n$ are the path, the cycle and the complete graph on $n$ vertices, respectively. As corollaries, we give sufficient conditions for these strong product graphs $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$ to be maximally restricted edge-connected and super restricted edge-connected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Restricted Edge-Connectivity of Strong Product Graphs Ye, Hazhe Tian, Yingzhi Combinatorics The restricted edge-connectivity of a connected graph $G$, denoted by $λ^{\prime}(G)$, if it exists, is the minimum cardinality of a set of edges whose deletion makes $G$ disconnected and each component with at least 2 vertices. It was proved that if $G$ is not a star and $|V(G)|\geq4$, then $λ^{\prime}(G)$ exists and $λ^{\prime}(G)\leqξ(G)$, where $ξ(G)$ is the minimum edge-degree of $G$. Thus a graph $G$ is called maximally restricted edge-connected if $λ^{\prime}(G)=ξ(G)$; and a graph $G$ is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of $G$. The strong product of graphs $G$ and $H$, denoted by $G\boxtimes H$, is the graph with vertex set $V(G)\times V(H)$ and edge set $\{(x_1,y_1)(x_2,y_2)\ |\ x_1=x_2$ and $y_1y_2\in E(H)$; or $y_1=y_2$ and $x_1x_2\in E(G)$; or $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$\}. In this paper, we determine, for any nontrivial connected graph $G$, the restricted edge-connectivity of $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$, where $P_n$, $C_n$ and $K_n$ are the path, the cycle and the complete graph on $n$ vertices, respectively. As corollaries, we give sufficient conditions for these strong product graphs $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$ to be maximally restricted edge-connected and super restricted edge-connected. |
| title | The Restricted Edge-Connectivity of Strong Product Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.15549 |