Let \(G\) be a graph with vertex set \(V(G)\) and let \(f\) be a nonnegative integer-valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called an \(f\)-factor if \(d_F(x) = f(x)\) for every \(x \in V(F)\). In this paper, we present some sufficient conditions for the existence of \(f\)-factors and connected \((f-2, f)\)-factors in \(K_{1,n}\)-free graphs. The conditions involve the minimum degree, the stability number, and the connectivity of graph \(G\).

We classify the minimal blocking sets of size 15 in \(\mathrm{PG}(2,9)\). We show that the only examples are the projective triangle and the sporadic example arising from the secants to the unique complete 6-arc in \(\mathrm{PG}(2,9)\). This classification was used to solve the open problem of the existence of maximal partial spreads of size 76 in \(\mathrm{PG}(3,9)\). No such maximal partial spreads exist \([13]\). In \([14]\), also the non-existence of maximal partial spreads of size 75 in \(\mathrm{PG}(3,9)\) has been proven. So, the result presented here contributes to the proof that the largest maximal partial spreads in \(\mathrm{PG}(3,q=9)\) have size \(q^2-q+2=74\).

Our work in this paper is concerned with a new kind of fuzzy ideal of a \(K\)-algebra called an \((\in, \in \vee_q)\)-fuzzy ideal. We investigate some interesting properties of \((\in, \in \vee_q)\)-fuzzy ideals of \(K\)-algebras. We study fuzzy ideals with thresholds which is a generalization of both fuzzy ideals and \((\in, \in \vee_q)\)-fuzzy ideals. We also present characterization theorems of implication-based fuzzy ideals.

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.