Let \(G\) be a connected graph. For \(S \subseteq V(G)\), the geodetic closure \(I_G[S]\) of \(S\) is the set of all vertices on geodesics (shortest paths) between two vertices of \(S\). We select vertices of \(G\) sequentially as follows: Select a vertex \(v_1\) and let \(S_1 = \{v_1\}\). Select a vertex \(v_2 \neq v_1\) and let \(S_2 = \{v_1, v_2\}\). Then successively select vertex \(v_i \notin I_G[S_{i-1}]\) and let \(S_i = \{v_1, v_2, \ldots, v_i\}\). We define the closed geodetic number (resp. upper closed geodetic number) of \(G\), denoted \(cgn(G)\) (resp. \(ucgn(G)\)), to be the smallest (resp. largest) \(k\) whose selection of \(v_1, v_2, \ldots, v_k\) in the given manner yields \(I_G[S_k] = V(G)\). In this paper, we show that for every pair \(a, b\) of positive integers with \(2 \leq a \leq b\), there always exists a connected graph \(G\) such that \(cgn(G) = a\) and \(ucgn(G) = b\), and if \(a < b\), the minimum order of such graph \(G\) is \(b\). We characterize those connected graphs \(G\) with the property: If \(cgn(G) < k < ucgn(G) = 6\), then there is a selection of vertices \(v_1, v_2, \ldots, v_k\) as in the above manner such that \(I_G[S_k] = V(G)\). We also determine the closed and upper closed geodetic numbers of some special graphs and the joins of connected graphs.

Let \(G\) be a graph with \(n\) vertices and suppose that for each vertex \(v\) in \(G\), there exists a list of \(k\) colors, \(L(v)\), such that there is a unique proper coloring for \(G\) from this collection of lists, then \(G\) is called a uniquely \(k\)-list colorable graph. We say that a graph \(G\) has the property \(M(k)\) if and only if it is not uniquely \(k\)-list colorable. M. Ghebleh and E. S. Mahmoodian characterized uniquely \(3\)-list colorable complete multipartite graphs except for the graphs \(K_{1*4,5}\), \(K_{1*5,4}, K_{1*4,4}\), \(K_{2,3,4}\), and \(K_{2,2,r}\), \(4 \leq r \leq 8\). In this paper, we prove that the graphs \(K_{1*4,5}\), \(K_{1*5,4}\), \(K_{1*4,4}\), and \(K_{2,3,4}\) have the property \(M(3)\).

Let \(G\) be a simple graph and \(f: V(G) \mapsto \{1, 3, 5, \ldots\}\) an odd integer valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a \((1, f)\)-odd factor if \(d_F(v) \in \{1, 3, \ldots, f(v)\}\) for all \(v \in V(G)\), where \(d_F(v)\) is the degree of \(v\) in \(F\). For an odd integer \(k\), if \(f(v) = k\) for all \(v\), then a \((1, f)\)-odd factor is called a \([1, k]\)-odd factor. In this paper, the structure and properties of a graph with a unique \((1, f)\)-odd factor is investigated, and the maximum number of edges in a graph of the given order which has a unique \([1, k]\)-odd factor is determined.

Erdős and Soifer \([3]\) and later Campbell and Staton \([1]\) considered a problem which was a favorite of Erdős \([2]\): Let \(S\) be a unit square. Inscribe \(n\) squares with no common interior point. Denote by \(\{e_1, e_2, \ldots, e_n\}\) the side lengths of these squares. Put \(f(n) = \max \sum\limits_{i=1}^n e_i\). And they discussed the bounds for \(f(n)\). In this paper, we consider its dual problem – covering a unit square with squares.

The well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph \(G\) in terms of the deficiency \(\max_{X \subseteq V(G)} \{ \omega_0(G – X) – |X| \}\) of \(G\), where \(\omega_0(H)\) denotes the number of odd components of \(H\). Let \(G’\) be the graph formed from \(G\) by subdividing (possibly repeatedly) a number of its edges. In this note we study the effect such subdivisions have on the difference between the size of a maximum matching in \(G\) and the size of a maximum matching in \(G’\).

In this paper, we give some necessary conditions for a prime graph. We also present some new families of prime graphs such as \(K_n \odot K_1\) is prime if and only if \(n \leq 7\), \(K_n \odot \overline{K_2}\) is prime if and only if \(n \leq 16\), and \(K_{m}\bigcup S_n\) is prime if and only if \(\pi(m+n-1) \geq m\). We also show that a prime graph of order greater than or equal to \(20\) has a nonprime complement.

Consider a lottery scheme consisting of randomly selecting a winning \(t\)-set from a universal \(m\)-set, while a player participates in the scheme by purchasing a playing set of any number of \(n\)-sets from the universal set prior to the draw, and is awarded a prize if \(k\) or more elements of the winning \(t\)-set occur in at least one of the player’s \(n\)-sets (\(1 \leq k \leq \{n,t\} \leq m\)). This is called a \(k\)-prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a \(k\)-prize, no matter which winning \(t\)-set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by \(L(m,n,t;k)\), and the number of such non-isomorphic sets is called the lottery characterisation number, denoted by \(\eta(m,n,t;k)\). In this paper, an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges \(2 \leq k \leq 4\), \(k \leq t \leq 11\), \(k \leq n \leq 12\), and \(\max\{n,t\} \leq m \leq 20\). In the process, \(32\) new lottery numbers are found, and bounds on a further \(31\) lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.

Beck’s coloring is studied for meet-semilattices with \(0\). It is shown that for such semilattices, the chromatic number equals the clique number.

The main result of this paper is an upper bound on the number of independent sets in a tree in terms of the order and diameter of the tree. This new upper bound is a refinement of the bound given by Prodinger and Tichy [Fibonacci Q., \(20 (1982), no. 1, 16-21]\). Finally, we give a sufficient condition for the new upper bound to be better than the upper bound given by Brigham, Chandrasekharan and Dutton [Fibonacci Q., \(31 (1993), no. 2, 98-104]\).

In this paper, it is shown that every extended directed triple system of order \(v\) can be embedded in an extended directed triple system of order \(n\) for all \(n \geq 2v\). This produces a generalization of the Doyen- Wilson theorem for extended directed triple systems.