arXiv subjects

Shu-Li Zhao

Publications and source records attributed to Shu-Li Zhao.

6 recordsLinked to original sources

arXiv2026

Upper bound on the size of subgraphs avoiding cycles of girth length in graphs

A graph $G$ is said to be $H$-free if it contains no copy of $H$. The generalized Turán number $\mathrm{ex}(G, H)$, a natural extension of the classical Turán number, is defined as the maximum number of edges in an $H$-free spanning subgraph of $G$. Although $\mathrm{ex}(G, H)$ has been extensively studied for various specific graphs $G$, such as hypercubes, complete transposition graphs, and doubled Johnson graphs, very few results are known for general graphs. For a graph $G$ of girth $g$, let $C_{g}$ denote the cycle of length $g$. In this paper, we investigate the generalized Turán number $\mathrm{ex}(G, C_{g})$ for graphs $G$ in which every edge is contained in a fixed number of $g$-cycles, and establish a general upper bound on it. This bound generalizes the classical results on hypercubes and provides a unified approach for general graphs. Furthermore, we apply this bound to derive explicit upper bounds for several specific graphs, including $K_{n,n,\ldots,n}$, $K_{n}\times K_{n}$, $C_{m}\times C_{n}$, $S_{n}$, $AG_{n}$, $CT_{n}$ and $J(n;k,k+1)$.

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arXiv2026

Extremal even-cycle-free subgraphs of the $k$-ary $n$-cube

A graph is said to be $H$-free if it contains no subgraph isomorphic to $H$. The generalized Turán number $\mathrm{ex}(G, H)$, a natural extension of the classical Turán number, is defined as the maximum number of edges in an $H$-free spanning subgraph of $G$. For any integer $\ell\geq 2$, let $C_{2\ell}$ denote the even cycle of length $2\ell$. So far, extensive studies of the generalized Turán number for the hypercube $Q_{n}$ have been conducted. The $k$-ary $n$-cube is an important family of graphs with many favorable properties, such as regularity and vertex-transitivity. It generalizes the hypercube and reduces to the hypercube when $k=2$. In this paper, we investigate the generalized Turán number $\mathrm{ex}(Q^{k}_{n}, C_{2\ell})$ of $Q^{k}_{n}$ for all even integers $k\geq 4$ and integers $\ell\geq 2$, generalizing the result for $Q_{n}$ [J. Graph Theory 16 (1992) 273-286] from $k=2$ to $k\geq 4$.

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arXiv2018

The $g$-good neighbour diagnosability of hierarchical cubic networks

Let $G=(V, E)$ be a connected graph, a subset $S\subseteq V(G)$ is called an $R^{g}$-vertex-cut of $G$ if $G-F$ is disconnected and any vertex in $G-F$ has at least $g$ neighbours in $G-F$. The $R^{g}$-vertex-connectivity is the size of the minimum $R^{g}$-vertex-cut and denoted by $κ^{g}(G)$. Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The $g$-good-neighbor diagnosability such that every fault-free node has at least $g$ fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the $g$-good-neighbor diagnosability of the hierarchical cubic networks $HCN_{n}$ under the PMC model for $1\leq g\leq n-1$ and the $MM^{*}$ model for $1\leq g\leq n-1$ is $2^{g}(n+2-g)-1$, respectively.

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arXiv2018

The generalized connectivity of some regular graphs

The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we study the generalized $3$-connectivity of some general $m$-regular and $m$-connected graphs $G_{n}$ constructed recursively and obtain that $κ_{3}(G_{n})=m-1$, which attains the upper bound of $κ_{3}(G)$ [Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for $G=G_{n}$. As applications of the main result, the generalized $3$-connectivity of many famous networks such as the alternating group graph $AG_{n}$, the $k$-ary $n$-cube $Q_{n}^{k}$, the split-star network $S_{n}^{2}$ and the bubble-sort-star graph $BS_{n}$ etc. can be obtained directly.

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arXiv2018

The generalized connectivity of $(n,k)$-bubble-sort graphs

Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2, \cdots, T_r$ in $G$ such that $V(T_i)\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. The generalized $k$-connectivity is a generalization of the traditional connectivity. In this paper, the generalized $3$-connectivity of the $(n,k)$-bubble-sort graph $B_{n,k}$ is studied for $2\leq k\leq n-1$. By proposing an algorithm to construct $n-1$ internally disjoint paths in $B_{n-1,k-1}$, we show that $κ_{3}(B_{n,k})=n-2$ for $2\leq k\leq n-1$, which generalizes the known result about the bubble-sort graph $B_{n}$ [Applied Mathematics and Computation 274 (2016) 41-46] given by Li $et$ $al.$, as the bubble-sort graph $B_{n}$ is the special $(n,k)$-bubble-sort graph for $k=n-1$.

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arXiv2018

Two kinds of generalized connectivity of dual cubes

Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $k$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{k}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, k\}$ and $i\neq j$. For an integer $r$ with $2\leq r\leq n$, the {\em generalized $r$-connectivity} of a graph $G$ is defined as $κ_{r}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$. The $r$-component connectivity $cκ_{r}(G)$ of a non-complete graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. These two parameters are both generalizations of traditional connectivity. Except hypercubes and complete bipartite graphs, almost all known $κ_{r}(G)$ are about $r=3$. In this paper, we focus on $κ_{4}(D_{n})$ of dual cube $D_{n}$. We first show that $κ_{4}(D_{n})=n-1$ for $n\geq 4$. As a corollary, we obtain $κ_{3}(D_{n})=n-1$ for $n\geq 4$. Furthermore, we show that $cκ_{r+1}(D_{n})=rn-\frac{r(r+1)}{2}+1$ for $n\geq 2$ and $1\leq r \leq n-1$.

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