Related papers: An equality for balanced digraphs

We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on $n$ vertices if and only if…

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every…

We introduce the cycle intersection graph of a graph, an adaptation of the cycle graph of a graph, and use the structure of these graphs to prove an upper bound for the decycling number of all even graphs. This bound is shown to be…

This note provides a complete solution to a certain version of the edge-isoperimetric problem for powers of a cycle graph. Namely, it shows that the maximum number of edges inside a vertex subset of $C_n^s$ of size $k$ is achieved by a set…

A descent of a labeled acyclic digraph is a directed edge $x\to y$ with $x>y$. In this paper, we find a recurrence for the number of labeled acyclic digraphs with a given number of descents.

The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no…

We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with…

We show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs that does not contain any of three forbidden digraphs, then $D$ contains every antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of $K_{2,…

A special case of a conjecture by Thomass\'e is that any oriented graph with minimum outdegree k contains a dipath of length 2k. For the sake of proving whether or not a counterexample exists, we present reductions and establish bounds on…

Let $D=(V(D), A(D))$ be a digraph of order $n$ and let $S\subseteq V(D)$ with $2\leq |S|\leq n$. A directed cycle $C$ of $D$ is called a directed $S$-Steiner cycle (or, an $S$-cycle for short) if $S\subseteq V(C)$. Steiner cycles have…

A new efficient algorithm is presented for finding all simple cycles that satisfy a length constraint in a directed graph. When the number of vertices is non-trivial, most cycle-finding problems are of practical interest for sparse graphs…

A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that…

Isaak posed the following problem. Suppose $T$ is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in $T$ equals the…

A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

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…

The feedback set problems are about removing the minimum number of vertices or edges from a graph to break all its cycles. Much effort has gone into understanding their complexity on planar graphs as well as on graphs of bounded degree. We…

We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of…

A cut in a digraph $D=(V,A)$ is a set of arcs $\{uv \in A: u\in U, v\notin U\}$, for some $U\subseteq V$. It is known that the arc set $A$ is covered by $k$ cuts if and only if it admits a $k$-coloring such that no two consecutive arcs $uv,…

For any directed graph G with vertex set V, the graph G^(d) is said to be a subset power of G and is defined to have vertex set equal to the set of d-element subsets of V; in G^(d), there is an edge from A to B if and only if we can label…