Related papers: Intersecting 1-factors and nowhere-zero 5-flows

A triangle-path in a graph $G$ is a sequence of distinct triangles $T_1,T_2,\ldots,T_m$ in $G$ such that for any $i, j$ with $1\leq i < j \leq m$, $|E(T_i)\cap E(T_{i+1})|=1$ and $E(T_i)\cap E(T_j)=\emptyset$ if $j > i+1$. A connected graph…

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\cdot\omega(G)$…

We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these…

Using Razborov's flag algebras we show that a triangle-free graph on n vertices contains at most (n/5)^5 cycles of length five. It settles in the affirmative a conjecture of Erdos.

A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\varphi:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored…

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ…

We prove that every 3-edge-connected graph $G$ has a 3-flow $\phi$ with the property that $|\mathop{supp}(\phi)| \ge \frac{5}{6} |E(G)|$. The graph $K_4$ demonstrates that this $\frac{5}{6}$ ratio is best possible; there is an infinite…

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau}…

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained…

Given a zero-sum function $\beta : V(G) \rightarrow \mathbb{Z}_3$ with $\sum_{v\in V(G)}\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \beta(v)$ in $\mathbb{Z}_3$ for every vertex $v\in V(G)$ is called a…

Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \rho(G) be the set of all primes which divide some character degree of G. The prime graph \Delta(G) attached to G is a graph whose vertex…

Minimal prime graphs are connected graphs on at least two vertices whose complements satisfy the following conditions: triangle-freeness, 3-colorability, and edge-maximality with respect to the latter two properties. These graphs are prime…

Petersen's seminal work in 1891 asserts that the edge-set of a cubic graph can be covered by distinct perfect matchings if and only if it is bridgeless. Actually, it is known that for a very large fraction of bridgeless cubic graphs, every…

A graph $G$ is $H$-free if it has no induced subgraph isomorphic to $H$, where $H$ is a graph. In this paper, we show that every $\frac{3}{2}$-tough $(P_4 \cup P_{10})$-free graph has a 2-factor. The toughness condition of this result is…

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a…

In this paper, we show that if a graph $G$ satisfies $c_{1}(G-X)+\frac{2}{3}c_{3}(G-X)\leq \frac{4}{3}|X|+\frac{1}{3}$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{5}\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of…

The neighbor connectivity of a graph $G$ is the least number of vertices such that removing their closed neighborhoods from $G$ results in a graph that is disconnected, complete or empty. If a~graph is used to model the topology of an…

The Erdos-Hajnal conjecture says that for every graph $H$ there exists $c>0$ such that $\max(\alpha(G),\omega(G))\ge n^c$ for every $H$-free graph $G$ with $n$ vertices, and this is still open when $H=C_5$. Until now the best bound known on…

A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that, for every pair of adjacent edges $e$ and $f$, $|\phi(e)| \neq |\phi(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we…

We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically…