Searcharxiv⌕ Search

arXiv subjects

Shuai Kou

Publications and source records attributed to Shuai Kou.

6 recordsLinked to original sources
arXiv2025

A degree sum condition for the existence of a quasi 5-contractible edge in a quasi 5-connected graph

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. We show that every 5-connected graph contains a quasi 5-contractible edge. Furthermore, we prove that a quasi 5-connected graph possesses a quasi 5-contractible edge, if the degree sum of any two vertices with distance at most two is at least 9. This result strengthens a theorem proved by Kriesell when $k=4$ (M. Kriesell, A degree sum condition for the existence of a contractible edge in a $k$-connected graph, J. Combin. Theory Ser. B 82(2001)81-101).

math.CO↗
arXiv2025

A forbidden pair for quasi 5-contractible edges

An edge of a quasi $k$-connected graph is said to be quasi $k$-contractible if the contraction of the edge results in a quasi $k$-connected graph. If every quasi $k$-connected graph without a quasi $k$-contractible edge has either $H_{1}$ or $H_{2}$ as a subgraph, then an unordered pair of graphs $\{H_{1}, H_{2}\}$ is said to be a forbidden pair for quasi $k$-contractible edges. We prove that $\{K_{4}^{-}, \overline{P_{5}}\}$ is a forbidden pair for quasi 5-contractible edges, where $K_{4}^{-}$ is the graph obtained from $K_{4}$ by removing just one edge and $\overline{P_{5}}$ is the complement of a path on five vertices.

math.CO↗
arXiv2025

A constructive characterization of uniformly 4-connected graphs

A constructive characterization of the class of uniformly $4$-connected graphs is presented. The characterization is based on the application of graph operations to appropriate vertex and edge sets in uniformly $4$-connected graphs, that is, any uniformly $4$-connected graph can be obtained from $C_5^2$ or $C_6^2$ by a number of $\Delta_1^+$ or $\Delta_2^+$-operations to quasi $4$-compatible sets.

math.CO↗
arXiv2025

Contractible Non-Edges in 3-Connected Graphs

We call a pair of non-adjacent vertices in G a non-edge. Contraction of a non-edge {u, v} in G is the replacement of u and v with a single vertex z and then making all the vertices that are adjacent to u or v adjacent to z. A non-edge {u, v} is said to be contractible in a k-connected graph G, if the resulting graph after its contraction remains k-connected. Tsz Lung Chan characterized all 3-connected graphs (finite or infinite) that does not contain any contractible non-edges in 2019, and posed the problem of characterizing all 3-connected graphs that contain exactly one contractible non-edge. In this paper, we solve this problem.

math.CO↗
arXiv2021

A note on $Oct_{1}^{+}$-free graphs and $Oct_{2}^{+}$-free graphs

Let $Oct_{1}^{+}$ and $Oct_{2}^{+}$ be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For $Oct_{1}^{+}$, we prove that a 4-connected graph is $Oct_{1}^{+}$-free if and only if it is $C_{6}^{2}$, $C_{2k+1}^{2}$ $(k \geq 2)$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices. We also show that a planar graph is $Oct_{1}^{+}$-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from $\{K_{1}, K_{2} ,K_{3}\} \cup \mathscr{K} \cup \{Oct,L_{5} \}$, where $\mathscr{K}$ is the set of graphs obtained by repeatedly taking the special 3-sums of $K_{4}$. For $Oct_{2}^{+}$, we prove that a 4-connected graph is $Oct_{2}^{+}$-free if and only if it is planar, $C_{2k+1}^{2}$ $(k \geq 2)$, $L(K_{3,3})$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices.

math.CO↗