4-Factor-Criticality of Vertex-Transitive Graphs

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G of order n is said to be k-factor-critical for integers $$1\le k < n$$ , if the removal of any k vertices results in a graph with a perfect matching. 1- and 2-factor-critical graphs are the well-known factor-critical and bicritical graphs, respectively. A k-factor-critical graph G is called minimal if for any edge $$e\in E(G)$$ , $$G-e$$ is not k-factor-critical. In 1998, O. Favaron and M. Shi conjectured that every minimally k-factor-critical graph of order n has the minimum degree $$k+1$$ and confirmed it for $$k=1, n-2, n-4$$ and $$n-6$$ . In this paper, we use a simple method to reprove the above results. As a main result, the further use of this method enables us to prove the conjecture to be true for $$k=n-8$$ . We also obtain that every minimally $$(n-6)$$ -factor-critical graph of order n has at most $$n-\Delta (G)$$ vertices with the maximum degree $$\Delta (G)$$ for $$\Delta (G)\ge n-4$$ .

A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $\mathrm{Aut}(\Gamma \times K_2)$ is not isomorphic to $\mathrm{Aut}(\Gamma) \times \mathbb{Z}_2$. We show that a connected and non-bipartite Cayley graph $\mathrm{Cay}(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ satisfying certain properties. The S-rings with these properties are characterized if $H$ is a cyclic group of twice odd order. As an application, a necessary and sufficient condition is given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$.

Chromatic-index-critical graphs of orders 13 and 14

A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t&lt;v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not $3$-extendable. We also show that strongly regular graphs of valency $k\geq 3$ with $\lambda \geq 1$ are $\lfloor k/3\rfloor$-extendable (when $\mu \leq k/2$) and $\lceil \frac{k+1}{4}\rceil$-extendable (when $\mu&gt;k/2$), where $\lambda$ is the number of common neighbors of any two adjacent vertices and $\mu$ is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best possible as there are strongly regular graphs of valency $k$ that are not $\lceil k/2\rceil $-extendable. We show that the extendability of many strongly regular graphs of valency $k$ is at least $\lceil k/2 \rceil -1$ and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order.

In this paper it is shown that if a graph G possesses a spanning subgraph H with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then G also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order n, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order n connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Bounds on the paired domination number of graphs with minimum degree at least three

Factor-critical property in 3-dominating-critical graphs

Homomorphisms of binary Cayley graphs

The structure of totally critical graphs of odd order

Vertex-magic labelings of regular graphs II

Two-arc-transitive graphs of odd order — II

Gossiping and routing in second-kind Frobenius graphs

Automorphisms of cubic Cayley graphs of order [formula omitted

On 4-ordered 3-regular graphs