Related papers: 3-Factor-criticality of vertex-transitive graphs

A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more…

A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least…

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…

We show that there exist functions $c$ and $g$ such that, if $k$, $n$ and $d$ are positive integers with $d> g(n)$ and $\Gamma$ is a $d$-valent $2$-arc-transitive graph of order $kp^n$ with $p$ a prime, then $p\leqslant kc(d)$. In other…

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and…

We prove that an eulerian graph $G$ admits a decomposition into $k$ closed trails of odd length if and only if and it contains at least $k$ pairwise edge-disjoint odd circuits and $k\equiv |E(G)|\pmod{2}$. We conjecture that a connected…

A bridgeless graph $G$ is called $3$-flow-critical if it does not admit a nowhere-zero $3$-flow, but $G/e$ has for any $e\in E(G)$. Tutte's $3$-flow conjecture can be equivalently stated as that every $3$-flow-critical graph contains a…

It is proved that for a graph of order $n$, where $n\ge 6$, if the signless Laplacian index is larger than or equal to certain value depending on $n$, then the graph contains a trebly chorded cycle, where the chords incident to a common…

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum…

In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\emptyset$, $\{x\}$ $(x\in V )$ and $V$ are modules of $G$, called trivial modules. A graph,…

We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…

For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices.…

For a nonnegative integer $k$, a graph $G$ is said to be $k$-factor-critical if $G-Q$ admits a perfect matching for any $Q\subseteq V(G)$ with $|Q|=k$. In this article, we prove spectral radius conditions for the existence of…

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$…

In this paper, we are concerned with sufficient conditions for the existence of a $\{P_{2},P_{2k+1}\}$-factor. We prove that for $k\geq 3$, there exists $\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\sum_{0\leq j\leq…

We prove that every internally 4-connected non-planar bipartite graph has an odd K_3,3 subdivision; that is, a subgraph obtained from K_3,3 by replacing its edges by internally disjoint odd paths with the same ends. The proof gives rise to…

We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the…

In this paper, we give the following result: If $D$ is a digraph of order $n$, and if $d_{D}^{+}(u) + d_{D}^{-}(v) \ge n$ for every two distinct vertices $u$ and $v$ with $(u, v) \notin A(D)$, then $D$ has a directed $2$-factor with exactly…

A graph $G$ is called $3$-choice critical if $G$ is not $2$-choosable but any proper subgraph is $2$-choosable. A characterization of $3$-choice critical graphs was given by Voigt in [On list Colourings and Choosability of Graphs,…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…