A sequence \( S \) is potentially \( K_{m}-C_4 \)-graphical if it has a realization containing a \( K_m-C_4 \) as a subgraph. Let \( \sigma(K_m-C_4,n) \) denote the smallest degree sum such that every \( n \)-term graphical sequence \( S \) with \( \sigma(S) \geq \sigma(K_m-C_4,n) \) is potentially \( K_m-C_4 \)-graphical. In this paper, we prove that \( \sigma(K_m-C_4,n) \geq (2m-6)n-(m-3)(m-2)+2 \), for \( n \geq m \geq 4 \). We conjecture that equality holds for \( n \geq m \geq 4 \). We prove that this conjecture is true for \( m = 5 \).

In 1975, Leech introduced the problem of finding trees whose edges can be labeled with positive integers in such a way that the set of distances (sums of weights) between vertices is \(\{1, 2, \dots, \binom{n}{2}\}\), where \(n\) is the number of vertices. We refer to such trees as perfect distance trees. More generally, we define a distinct distance tree to be a weighted tree in which the distances between vertices are distinct. In this article, we focus on identifying minimal distinct distance trees. These are the distinct distance trees on \(n\) vertices that minimize the maximum distance between vertices. We determine \(M(n)\), the maximum distance in a minimal distinct distance tree on \(n\) vertices, for \(n \leq 10\), and give bounds on \(M(n)\) for \(n \geq 11\). This includes a determination of all perfect distance trees for \(n < 18\). We then consider trees according to their diameter and show that there are no further perfect distance trees with diameter at most \(3\). Finally, generalizations to graphs, forests, and distinct distance sets are considered.

A bijection \( \lambda: V \cup E \cup F \to \{1, 2, 3, \dots, |V| + |E| + |F|\} \) is called a \( d \)-antimagic labeling of type \( (1, 1, 1) \) of plane graph \( G(V, E, F) \) if the set of \( s \)-sided face weights is \( W_s = \{a_s + a_s+d, a_s+2d, \dots, a_s + (f_s-1)d\} \) for some integers \( s \), \( a_s \), and \( d \), where \( f_s \) is the number of \( s \)-sided faces and the face weight is the sum of the labels carried by that face and the edges and vertices surrounding it. In this paper, we examine the existence of \( d \)-antimagic labelings of type \( (1, 1, 1) \) for a special class of plane graphs \( {C}_a^b \).

GWhD(\(v\))s, or Generalized Whist Tournament Designs on \( v \) players, are a relatively new type of design. GWhD(\(v\))s are (near) resolvable (\(v,k,k-1\)) BIBDs. For \( k = et \), each block of the design is considered to be a game involving \( e \) teams of \( t \) players each. The design is subject to the requirements that every pair of players appears together in the same game exactly \( t-1 \) times as teammates and exactly \( k-t \) times as opponents. These conditions are referred to as the Generalized Whist Conditions, and when met, we refer to the (N)RBIBD as a (\( t, k \)) GWhD(\(v\)). When \( k = 10 \), necessary conditions on \( v \) are that \( v \equiv 0, 1 \pmod{10} \). In this study, we focus on the existence of (\(2,10\)) GWhD(\(v\)), \(v \equiv 1 \pmod{10}\). It is known that a (\(2,10,9\))-NRBIBD does not exist. Therefore, it is impossible to have a (\(2,10\)) GWhD(\(21\)). It is established here that (\(2,10\)) GWhD(\(10n+1\)) exist for all other \(v\) with at most 42 additional possible exceptions.

We define an overfull set of one-factors of \( K_{2n} \) to be a set of one-factors that between them cover all the edges of \( K_{2n} \), but contain no one-factorization of \( K_{2n} \). We address the question: how many members can such a set contain?

In this paper, we shall consider acquisition sequences of a graph. The formation of each acquisition sequence is a process that creates an independent set. Each acquisition sequence is a sequence of “acquisitions” which are defined on a graph \( G \) for which each vertex originally has a value of one associated with it. In an acquisition, a vertex transfers all of its value to an adjacent vertex with equal or greater value. For an acquisition sequence, one continues until no more acquisitions are possible. The parameter \( a(G) \) is defined to be the minimum possible number of vertices with a nonzero value at the conclusion of such an acquisition sequence. Clearly, if \( S \) is a set of vertices with nonzero values at the end of some acquisition sequence, then \( S \) is independent, and we call such a set \( S \) an acquisition set. We show that for a given graph \( G \), “Is \( a(G) = 1 \)” is NP-complete, and describe a linear time algorithm to determine the acquisition number of a caterpillar.

The cardinality of the minimal pairwise balanced designs on \( v \) elements with largest block size \( k \) is denoted by \( g^{(k)}(v) \). It is known that \( 31 \leq g^{(4)}(18) \leq 33 \). In this paper, we show that \( g^{(4)}(18) \neq 31 \).

In 1966, Wagner used computational search methods to construct a \([23,14,5]\) code. This code has been examined with much interest since that time, in hopes of finding a geometric construction and possible code extensions. In this article, we give a simple geometric construction for Wagner’s code and consider extensions of this construction.

A graph \( G \) is said to be \( E_k \)-Cordial if there is an edge labeling \( f : E(G) \rightarrow \{0,1,\ldots,k-1\} \) such that, at each vertex \( v \), the sum modulo \( k \) of the labels on the edges incident with \( v \) is \( f(v) \) and it satisfies the inequalities \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \), where \( v_f(s) \) and \( e_f(t) \) are, respectively, the number of vertices labeled with \( s \) and the number of edges labeled with \( t \). The map \( f \) is then called an \( E_k \)-cordial labeling of \( G \).

This paper investigates \( E_3 \)-cordiality of snakes, one point unions, path unions, and coronas involving complete graphs.