If \(T\) is a tree on \(n\) vertices, \(n \geq 3\), and if \(G\) is a connected graph such that \(d(u) + d(v) + d(u,v) \geq 2n\) for every pair of distinct vertices of \(G\), it has been conjectured that \(G\) must have a non-separating copy of \(T\). In this note, we prove this result for the special case in which \(d(u) + d(v) + d(u,v) \geq 2n + 2\) for every pair of distinct vertices of \(G\), and improve this slightly for trees of diameter at least four and for some trees of diameter three.

Let \(G\) be a finite simple group, \(M\) be a maximal subgroup of \(G\) and \(C_g = nX\) be the conjugacy class of \(G\) containing \(g\). In this paper we discuss a new method for constructing \(1-(v,k,\lambda)\) designs \(\mathcal{D} = (\mathcal{P},\mathcal{B})\), where \(\mathcal{P} = nX\) and \(\mathcal{B} = \{(M\cap nX)^y \mid y \in G\}\). The parameters \(v\), \(k\), \(\lambda\) and further properties of \(\mathcal{D}\) are determined. We also study codes associated with these designs.

Let \(G\) be a graph and \(H\) a subgraph of \(G\). A \(D(G, H, \lambda)\) design is a collection \(\mathcal{D}\) of subgraphs of \(G\) each isomorphic to \(H\) so that every \(2\)-path (path of length \(2\)) in \(G\) lies in exactly \(\lambda\) subgraphs in \(\mathcal{D}\). The problem of constructing \(D(K_n,C_n,1)\) designs is the so-called Dudeney’s round table problem. We denote by \(C_k\), a cycle on \(k\) vertices and by \(P_k\), a path on \(k\) vertices.

In this paper, we construct \(D(K_{n,n},C_{2n},1)\) designs and \(D(K_n,P_n,1)\) designs when \(n \equiv 0,1,3 \pmod{4}\); and \(D(K_{n,n},C_{2n},2)\) designs and \(D(K_n,P_n,2)\) designs when \(n \equiv 2 \pmod{4}\). The existence problems of \(D(K_{n,n},C_{2n},1)\) designs and \(D(K_n,P_n,1)\) designs for \(n \equiv 2 \pmod{4}\) remain open.

The spread of a graph \(G\) is defined as the difference between the largest and smallest eigenvalues of \(G\). Using the lower bounds obtained by Liu and Liu in [4] on the spread of a graph, we in this note present spread conditions for some Hamiltonian properties of a graph.

A \({vertex \;cover}\) of a graph \(G = (V, E)\) is a subset \(S \subseteq V\) such that every edge is incident with at least one vertex in \(S\), and \(\alpha(G)\) is the cardinality of a smallest vertex cover. For a given vertex cover \(S\), a defense by \(S\) to an attack on an edge \(e = \{v, w\}\), where \(v \in S\), is a one-to-one function \(f : S \to V\), such that:

1. \(f(v) = w\), and
2. for each \(s \in S – v\), \(f(s) \in N[s]\).

Informally, a set is an \({eternal\; vertex \;cover}\) if it can defend an “attack” on any edge and the process can be repeated indefinitely. The cardinality of a smallest eternal vertex cover is denoted \(\alpha_{m}^\infty(G)\). A set of vertices which is not an eternal vertex cover is \({mortal}\). A formal definition of eternal vertex cover is provided and demonstrated to be equivalent to a characterization using closed families of vertex covers.
Eternal vertex covers are shown to be closed under taking supersets and a lower bound for \(\alpha_{m}^\infty(G)\) is given which depends on the vertex connectivity number and the independent domination number. A corresponding upper bound is given for the size of a mortal set. The \({death \;spiral\; number}\) of a mortal vertex cover is defined and used to partition the collection of all mortal sets. Mortal sets are shown to be closed under taking subsets implying the collection of mortal sets for a graph with at least one edge is an independence system. The death spiral number of a graph is the maximum of the death spiral numbers of all mortal sets.
An optimal attack/defense strategy is determined for a set of size \(\alpha_{m}^\infty(T) – 1\) in a tree \(T\), along with a polynomial labeling algorithm which computes its death spiral number.

We first introduce the concept of \((k, k’, k”)\)-domination numbers in graphs, which is a generalization of many domination parameters. Then we find lower and upper bounds for this parameter, which improve many well-known results in the literature.

We are interested in ordering the elements of a subset \( A \) of the non-zero integers modulo \( n \) in such a way that all the partial sums are distinct. We conjecture that this can always be done, and we prove various partial results about this problem.

A graph \( G \) with maximum degree \( \Delta \) and edge chromatic number \( \chi'(G) > \Delta \) is \emph{edge-\(\Delta\)-critical} if \( \chi'(G-e) = \Delta \) for each \( e \in E(G) \). In this article, we provide a new proof of adjacency Lemmas on edge-critical graphs such that Vizing’s adjacency lemma becomes a corollary of our results.

This paper surveys recent results for flag enumeration of polytopes, Bruhat graphs, balanced digraphs, Whitney stratified spaces and quasi-graded posets.

A bipancyclic graph on \( v \) vertices is a bipartite graph that contains, as subgraphs, cycles of length \( n \) for every even integer \( n \) such that \( 4 \leq n \leq v \). Such a graph is uniquely bipancyclic if it contains exactly one subgraph of each permissible length.

In this paper, we find all uniquely bipancyclic graphs on 30 or fewer vertices.