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

We prove that every connected cubic graph with $n$ vertices has a maximal matching of size at most $\frac{5}{12} n+ \frac{1}{2}$. This confirms the cubic case of a conjecture of Baste, F\"urst, Henning, Mohr and Rautenbach (2019) on regular…

A dynamic coloring of the vertices of a graph $G$ starts with an initial subset $S$ of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor…

We show that a cubic graph $G$ of order $n$ has an induced $2$-regular subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no induced cycle of…

A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges. Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this…

A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $\phi$ is an $A$-flow in $G$, then let $N(\phi)$ be the nullity of $\phi$, that is, the set of…

We call a set $\mathcal S$ of graphs an "even subdivison-factor" of a cubic graph $G$ if $G$ contains a spanning subgraph $H$ such that every component of $H$ has an even number of vertices and is a subdivision of an element of $\mathcal…

Let G=(V,E) be a simple undirected graph. For a given set L of the real line, a function omega from E to L is called an L-flow. Given a vector gamma whose coordinates are indexed by V, we say that omega is a gamma-L-flow if for each v in V,…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

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…

Let $G$ be a graph. The maximum nullity of $G$, denoted by $M(G)$, is defined to be the largest possible nullity over all real symmetric matrices $A$ whose $a_{ij}\neq 0$ for $i\neq j$, whenever two vertices $u_i$ and $u_j$ of $G$ are…

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. We denote by $f(G)$ and $F(G)$ the minimum and maximum forcing number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique…

A graph is $k$-vertex-critical if $\chi(G)=k$ but $\chi(G-v)<k$ for all $v\in V(G)$. We construct a new infinite families of $k$-vertex-critical $(P_5,C_5)$-free graphs for all $k\ge 6$. Our construction generalizes known constructions for…

Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the…

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring…

An edge-colored graph $G$ is rainbow connected if every pair of vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of $G$ is defined to be the minimum integer $t$ such that there…

Let $\Gamma$ be a graph, $A$ an abelian group, $\mathcal{D}$ a given orientation of $\Gamma$ and $R$ a unital subring of the endomorphism ring of $A$. It is shown that the set of all maps $\varphi$ from $E(\Gamma)$ to $A$ such that…

We prove that the clique graph operator $k$ is divergent on a locally cyclic graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $\delta(G)=6$ if and only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the maximal…

In this paper we study minimum cut and maximum flow problems on planar graphs, both in static and in dynamic settings. First, we present an algorithm that given an undirected planar graph computes the minimum cut between any two given…

Minimal prime graphs (MPGs) are a special class of prime graphs (also known as Gruenberg-Kegel graphs) associated with finite solvable groups. A graph is an MPG if it has at least two vertices, is connected, its complement is triangle-free…

If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a{\em Fulkerson covering}) with the property that every edge of $G$ is contained in exactly two of them. A consequence of the Fulkerson…