Related papers: Some results on Berge's conjecture and Begin-End c…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

Bollob\'{a}s and Scott [5] conjectured that every graph $G$ has a balanced bipartite spanning subgraph $H$ such that for each $v\in V(G)$, $d_H(v)\ge (d_G(v)-1)/2$. In this paper, we show that every graphic sequence has a realization for…

Recently, the authors gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number and…

For a graph $G$ with the vertex set $V(G)$ and the edge set $E(G)$ and a star subgraph $S$ of $G$, let $\alpha_S(G)$ be the maximum number of vertices in $G$ such that no two of them are in the same star subgraph $S$ and $\theta_S(G)$ be…

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this…

Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which…

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$…

Let $ D $ be a finite digraph, and let $ V_0,\dots,V_{k-1} $ be nonempty subsets of $ V(D) $. The (strong form of) Edmonds' branching theorem states thatthere are pairwise edge-disjoint spanning branchings $ \mathcal{B}_0,\dots,…

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…

A conjecture of Jackson from 1981 states that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian. We prove this conjecture for sufficiently large $n$. In fact we prove a more general result that for all…

A classical result by Erdos and Posa states that there is a function $f: {\mathbb N} \rightarrow {\mathbb N}$ such that for every $k$, every graph $G$ contains $k$ pairwise vertex disjoint cycles or a set $T$ of at most $f(k)$ vertices such…

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has…

Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of…

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let $\beta(G)$ denote the size of the smallest subset X in E(G) such that $G\X$ has no directed cycles, and let $\gamma(G)$ be the number…

A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such…

Let $P$ be a set of $n$ points in $\mathrm{R}^2$, and let $\mathrm{DT}(P)$ denote its Euclidean Delaunay triangulation. We introduce the notion of an edge of $\mathrm{DT}(P)$ being {\it stable}. Defined in terms of a parameter $\alpha>0$, a…

A parallel $d$-stable trace is a closed walk which traverses every edge of a graph exactly twice in the same direction and for every vertex $v$, there is no subset $X \subseteq N(v)$ with $1 \leq |N| \leq d$ such that every time the walk…

A subset $D\subseteq V_G$ is a dominating set of $G$ if every vertex in $V_G-D$ has a~neighbor in $D$, while $D$ is a paired-dominating set of $G$ if $D$ is a~dominating set and the subgraph induced by $D$ contains a perfect matching. A…

Given a (di)graph $H$, we say that a (di)graph $H^\prime$ is an $H$-subdivision if $H^\prime$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint path(s). Pavez-Sign\'{e} conjectured that for every…

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…