Let \(G\) be a digraph. For two vertices \(u\) and \(v\) in \(G\), the distance \(d(u,v)\) from \(u\) to \(v\) in \(G\) is the length of the shortest directed path from \(u\) to \(v\). The eccentricity \(e(v)\) of \(v\) is the maximum distance of \(v\) to any other vertex of \(G\). A vertex \(u\) is an eccentric vertex of \(v\) if the distance from \(v\) to \(u\) is equal to the eccentricity of \(v\). The eccentric digraph \(ED(G)\) of \(G\) is the digraph that has the same vertex set as \(G\) and the arc set defined by: there is an arc from \(u\) to \(v\) if and only if \(v\) is an eccentric vertex of \(u\). In this paper, we determine the eccentric digraphs of digraphs for various families of digraphs and we get some new results on the eccentric digraphs of the digraphs.

We present \(3\) open challenges in the field of Costas arrays. They are: a) the determination of the number of dots on the main diagonal of a Welch array, and especially the maximal such number for a Welch array of a given order; b) the conjecture that the fraction of Welch arrays without dots on the main diagonal behaves asymptotically as the fraction of permutations without fixed points and hence approaches \(1/e\) and c) the determination of the parity populations of Golomb arrays generated in fields of characteristic \(2\).

Let \(G\) be the graph obtained from \(K_{3,3}\) by deleting an edge. We find a list assignment with \(|L(v)| = 2\) for each vertex \(v\) of \(G\), such that \(G\) is uniquely \(L\)-colorable, and show that for any list assignment \(L’\) of \(G\), if \(|Z'(v)| \geq 2\) for all \(v \in V(G)\) and there exists a vertex \(v_0\) with \(|L'(v_0)| > 2\), then \(G\) is not uniquely \(L’\)-colorable. However, \(G\) is not \(2\)-choosable. This disproves a conjecture of Akbari, Mirrokni, and Sadjad (Problem \(404\) in Discrete Math. \(266(2003) 441-451)\).

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. In this note, we show that the vertex set of every graph with minimum degree at least two and with no component that is a \(5\)-cycle can be partitioned into a dominating set and a total dominating set.

Let \(G\) be an undirected graph, \(A\) be an (additive) Abelian group and \(A^* = A – \{0\}\). A graph \(G\) is \(A\)-connected if \(G\) has an orientation such that for every function \(b: V(G) \longmapsto A\) satisfying \(\sum_{v\in V(G)} b(v) = 0\), there is a function \(f: E(G) \longmapsto A^*\) such that at each vertex \(v\in V(G)\) the net flow out of \(v\) equals \(b(v)\). We investigate the group connectivity number \(\Lambda_g(G) = \min\{n; G \text{ is } A\text{-connected for every Abelian group with } |A| \geq n\}\) for complete bipartite graphs, chordal graphs, and biwheels.

Various enumeration problems for classes of simply generated families of trees have been the object of investigation in the past. We mention the enumeration of independent subsets, connected subsets or matchings for instance. The aim of this paper is to show how combinatorial problems of this type can also be solved for rooted trees and trees, which enables us to take better account of isomorphisms. As an example, we will determine the average number of independent vertex subsets of trees and binary rooted trees (every node has outdegree \(\leq 2\)).

In this paper, first we introduce the concept of a \({connected}\) graph homomorphism as a homomorphism for which the inverse image of any edge is either empty or a connected graph, and then we concentrate on chromatically connected (resp. chromatically disconnected) graphs such as \(G\) for which any \(\chi(G)\)-colouring is a connected (resp. disconnected) homomorphism to \(K_{\chi(G)}\).

In this regard, we consider the relationships of the new concept to some other notions as uniquely-colourability. Also, we specify some classes of chromatically disconnected graphs such as Kneser graphs \(KG(m,n)\) for which \(m\) is sufficiently larger than \(n\), and the line graphs of non-complete class II graphs.

Moreover, we prove that the existence problem for connected homomorphisms to any fixed complete graph is an NP-complete problem.

We show that every \(2\)-connected cubic graph of order \(n > 8\) admits a \(P_3\)-packing of at least \(\frac{9n}{11}n\) vertices. The proof is constructive, implying an \(O(M(n))\) time algorithm for constructing such a packing, where \(M(n)\) is the time complexity of the perfect matching problem for \(2\)-connected cubic graphs.

The locally twisted cube \(LTQ_n\) is a newly introduced interconnection network for parallel computing. As a variant of the hypercube \(Q_n\), \(LTQ_n\) has better properties than \(Q_n\) with the same number of links and processors. Yang, Megson and Evans Evans [Locally twisted cubes are \(4\)-pancyclic, Applied Mathematics Letters, \(17 (2004), 919-925]\) showed that \(LTQ_n\) contains a cycle of every length from \(4\) to \(2^n\). In this note, we improve this result by showing that every edge of \(LTQ_n\) lies on a cycle of every length from \(4\) to \(2^n\) inclusive.

Necessary and sufficient conditions are given for the existence of a \((K_3 + e, \lambda)\)-group divisible design of type \(g^tu^1\).