We give a necessary and sufficient condition of Hall’s type for a family of sets of even cardinality to be decomposable into two subfamilies having a common system of distinct representatives. An application of this result to partitions of Steiner Triple Systems into small configurations is presented.

In this paper, we construct 2-factorizations of \(K_n\) (\(n\) odd) containing a specified number, \(k\), of 6-cycles, for all integers \(k\) between 0 and the maximum possible expected number of 6-cycles in any 2-factorization, and for all odd \(n\), with no exceptions.

We deal with \((a,d)\)-face antimagic labelings of a certain class of plane quartic graphs. A connected plane graph \(G = (V, E, F)\) is said to be \((a,d)\)-\emph{face antimagic} if there exist positive integers \(a\) and \(d\), and a bijection \(g : E(G) \rightarrow \{1,2,…,|E(G)|\}\) such that the induced mapping \(\varphi_g : F(G) \rightarrow {N}\), defined by \(\varphi_g(f) = \sum\{g(e): e \in E(G) \text{ adjacent to face } f\}\), is injective and \(\varphi_g(F) = \{a,a+d,…,a+ (|F(G)| – 1)d\}\).

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\). A vertex labelling \(f : V \rightarrow \{0,1\}\) induces an edge labelling \(\overline{f} : E \rightarrow \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\). In this paper, we show that for every positive integer \(t\) and \(n\) the following families are cordial: (1) Helms \(H_{n}\). (2) Flower graphs \(FL_{n}\). (3) Gear graphs \(G_{n}\). (4) Sunflower graphs \(SFL_{n}\). (5) Closed helms \(CH_{n}\). (6) Generalised closed helms \(CH(t,n)\). (7) Generalised webs \(W(t, n)\).

A cycle \(C\) of a graph \(G\) is called a \(q\)-dominating cycle if every vertex of \(G\) which is not contained in \(C\) is adjacent to at least \(q\) vertices of \(C\). Let \(G\) be a \(k\)-connected graph with \(k \geq 2\). We present a sufficient condition, in terms of the degree sum of \(k + 1\) independent vertices, for \(G\) to have a \(qg\)-dominating cycle. This is an extension of a 1987 result by J.A. Bondy and G. Fan. Furthermore, examples will show that the given condition is best possible.

In an earlier paper [11], we proved that there does not exist any \(\Delta\)-critical graph of even order with five major vertices. In this paper, we prove that if \(G\) is a \(\Delta\)-critical graph of odd order \(2n+1\) with five major vertices, then \(e(G) = n\Delta+1\). This extends an earlier result of Chetwynd and Hilton, and also completes our characterization of graphs with five major vertices. In [9], we shall apply this result to establish some results on class 2 graphs whose core has maximum degree two.

In this paper, uniquely list colorable graphs are studied. A graph \(G\) is said to be uniquely \(k\)-list colorable if it admits a \(k\)-list assignment from which \(G\) has a unique list coloring. The minimum \(k\) for which \(G\) is not uniquely \(k\)-list colorable is called the \(m\)-number of \(G\). We show that every triangle-free uniquely colorable graph with chromatic number \(k+1\) is uniquely \(k\)-list colorable. A bound for the \(m\)-number of graphs is given, and using this bound it is shown that every planar graph has \(m\)-number at most \(4\). Also, we introduce list criticality in graphs and characterize all \(3\)-list critical graphs. It is conjectured that every \(\chi_\ell’\)-critical graph is \(\chi’\)-critical, and the equivalence of this conjecture to the well-known list coloring conjecture is shown.

A labeling \(f\) of the vertices of a graph \(G\) is said \(k\)-\emph{equitable} if each weight induced by \(f\) on the edges of \(G\) appears exactly \(k\) times. A graph \(G\) is said \emph{equitable} if for every proper divisor \(k\) of its size, the graph \(G\) has a \(k\)-equitable labeling.

A graph \(G\) is a corona graph if \(G\) is obtained from two graphs, \(G_1\) and \(G_2\), taking one copy of \(G_ 1\), which is supposed to have order \(p$, and \(p\) copies of \(G_2\), and then joining by an edge the \(k^{th}\) vertex of \(G_1\) to every vertex in the \(k^{th}\) copy of \(G_2\). We denote \(G\) by \(G_1 \otimes G_2\).

In this paper, we proved that the corona graph \(C_n \otimes K_1\) is equitable. Moreover, we show \(k\)-equitable labelings of the corona graph \(C_m \otimes nK_1\), for some values of the parameters \(k, m,\) and \(n\).

In this paper, we derive a necessary existence condition involving the parameters of a balanced array (B-array) with two symbols and of strength \(t = 8\). Consequently, we demonstrate that the existence condition derived here can provide us with useful information on the maximum number of constraints for B-arrays with a given number of columns.

A set of \(n+1\) orthogonal squares of order \(n\) is known to be equivalent to a complete set of \(n-1\) mutually orthogonal Latin squares of order \(n\) together with canonical row and column squares. In this note, we show that this equivalence does not extend to orthogonal hypercubes of dimensions \(d > 2\) by providing examples of affine designs that can be represented by complete sets of type \(0\) orthogonal hypercubes but not by complete sets of orthogonal Latin hypercubes together with canonical hypercubes that generalize the row and column squares in the case where \(d = 2\). These examples also clarify the relationship between affine designs and orthogonal hypercubes that generalize the classical equivalence between affine planes and complete sets of MOLS.

We conclude with the statement of a number of conjectures regarding some open questions.

We prove that if \(S\) is a quasiminimal generating set of a group \(\Gamma\) and \(F\) is an oriented forest with \(|S| > 2\) arcs, then the Cayley graph \({Cay}(\Gamma, S)\) can be decomposed into \(|\Gamma|\) arc-disjoint subdigraphs, each of which is isomorphic to \(F\).

The quantity \(g_2^{(k)}(v)\) is the minimum number of blocks in a family of blocks from a \(v\)-set that covers all \(\binom{v}{2}\) pairs exactly twice, given the restriction that the longest block in the covering family has length \(k\) (there may be many blocks of length \(k\)). We give certain results for the case \(k = 4\).

A set of edges \(D\) in a graph \(G\) is a dominating set of edges if every edge not in \(D\) is adjacent to at least one edge in \(D\). The minimum cardinality of an edge dominating set of \(G\) is the edge domination number of \(G\), denoted \(D_E(G)\). A graph \(G\) is edge domination critical, or \(EDC\), if for any vertex \(v\) in \(G\) we have \(D_E(G – v) = D_E(G) – 1\). Every graph \(G\) must have an induced subgraph \(F\) such that \(F\) is \(EDC\) and \(D_E(G) = D_E(F)\). In this paper, we prove that no tree with more than 2 vertices is \(EDC\), develop a forbidden subgraph characterization for the edge domination number of a tree, and we develop a construction that conserves the \(EDC\) property.

Let \(V\) be a finite set of order \(v\). A \((v,k,\lambda)\) covering design of index \(\lambda\) and block size \(k\) is a collection of \(k\)-element subsets, called blocks, such that every \(2\)-subset of \(V\) occurs in at least \(\lambda\) blocks. The covering problem is to determine the minimum number of blocks, \(\alpha(v, k, \lambda)$, in a covering design. It is well known that \(\alpha(v, k, \lambda) \geq \left\lceil\frac{v}{k} \lceil \frac{v-1}{k-1}.\lambda \rceil \right\rceil=\phi(v, k, \lambda)\), where \(\lceil x \rceil\) is the smallest integer satisfying \(x\leq\lceil x \rceil\). In this paper, we determine the value \(\alpha(v,5,\lambda)\), with few possible exceptions, for \(\lambda = 3\), \(v \equiv 2 \pmod{4}\) and \(\lambda = 9, 10, v\geq5\), and \(\lambda \geq 11\), \(v \equiv 2 \pmod{4}\).

Let \(G = (V, E)\) be a connected undirected graph. Suppose a fire breaks out at a vertex of \(G\) and spreads to all its unprotected neighbours in each time interval. Also, one vertex can be protected in each time interval. We are interested in the number of vertices that can be “saved”, that is, which will never be burned. An algorithm is presented to find the optimal solution in the 2-dimensional grid graphs and 3-dimensional cubic graphs. We also determined the upper and lower bounds of the maximum number of vertices that can be saved on the large product graphs. The problem of containing the fire with one firefighter or more is also considered.

Let \(C\) be the underlying graph of a configuration of \(l\) blocks in a path design of order \(v\) and block size \(3\), \((V, \mathcal{B})\). We say that \((V, \mathcal{B})\) is \((l,C)\)-ordered if it is possible to order its blocks in such a way that each set of \(l\) consecutive blocks has the same underlying graph \(C\). In this paper, we completely solve the problem of the existence of a \((2,C)\)-ordered path design \(P(v, 3, 1)\) for any configuration having two blocks.

Summary. In this paper, we present some inequalities on balanced arrays ({B-arrays}) of strength five with two symbols.