In this paper, we provide bounds for the crossing number of mesh connected trees and 3-regular mesh of trees.

The ordinary factorial may be written in terms of the Stirling numbers of the second kind as shown by Quaintance and Gould and the odd double factorial in terms of the Stirling numbers of the first kind as shown by Callan. During the preparation of an expository paper on Wolstenholme’s Theorem, we discovered an expression for the odd double factorial using the Stirling numbers of the second kind. This appears to be the first such identity involving both positive and negative integers and the result is outlined here.

Let \( G \) be a \( k \)-connected (\( k > 2 \)) graph of order \( n \). If \( \chi(G) > n-k \), then \( G \) is Hamiltonian or \( K_k \vee (K_{n-2k}) \) with \( n > 2k+1 \), where \( \chi(G) \) is the chromatic number of the graph \( G \).

An \((n, r)\)-arc in \(PG(2, q)\) is a set of \(n\) points such that each line contains at most \(r\) of the selected points. We show that in the case of the existence of a \((101, 10)\)-arc in \(PG(2, 11)\) it only admits the trivial linear automorphism.

In this paper, we generalise the notion of distance irregular labeling introduced by Slamin to vertex irregular \(d\)-distance vertex labeling, for any distance \(d\) up to the diameter. We also define the inclusive vertex irregular \(d\)-distance vertex labeling. We give the lower bound of the inclusive vertex irregular \(1\)-distance vertex labeling for general graphs and a better lower bound on caterpillars. The inclusive labelings for paths \(P_n, n \equiv 0 \mod 3\), stars \(S_n\), double stars \(S(m,n)\), cycles \(C_n\), and wheels \(W_n\) are provided. From the inclusive vertex irregular \(1\)-distance vertex labeling on cycles, we derive the vertex irregular \(1\)-distance vertex labeling on prisms.

A Hamiltonian walk in a nontrivial connected graph \( G \) is a closed walk of minimum length that contains every vertex of \( G \). The 3-path graph \( P_3(G) \) of a connected graph \( G \) of order 3 or more has the set of all 3-paths (paths of order 3) of \( G \) as its vertex set and two vertices of \( P_3(G) \) are adjacent if they have a 2-path in common. With the aid of Hamiltonian walks in spanning trees of the 3-path graph of a graph, it is shown that if \( G \) is a connected graph with minimum degree at least 4, then \( P_3(G) \) is Hamiltonian-connected.

The paired domination subdivision number \( sd_p(G) \) of a graph \( G \) is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of \( G \). We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the paired domination multisubdivision number of a nonempty graph \( G \), denoted by \( msd_p(G) \), as the minimum positive integer \( k \) such that there exists an edge which must be subdivided \( k \) times to increase the paired domination number of \( G \). We show that \( msd_p(G) \leq 4 \) for any graph \( G \) with at least one edge. We also determine paired domination multisubdivision numbers for some classes of graphs. Moreover, we give a constructive characterizations of all trees with paired domination multisubdivision number equal to 4.

Given a permutation \( \pi = (\pi_1, \pi_2, \pi_3, \ldots, \pi_n) \) over the alphabet \(\Sigma = \{0, 1, \ldots, n-1\}\), \(\pi_i\) and \(\pi_{i+1}\) are said to form an adjacency if \(\pi_{i+1} = \pi_i + 1\) where \(1 \leq i \leq n-1\). The set of permutations over \(\Sigma\) is a symmetric group denoted by \(S_n\). \(S_n(k)\) denotes the subset of permutations with exactly \(k\) adjacencies. We study four adjacency types and efficiently compute the cardinalities of \(S_n(k)\). That is, we compute for all \(k\) \(|S_n(k)|\) for each type of adjacency in \(O(n^2)\) time. We define reduction and show that \(S_n(n-k)\) is a multiset consisting exclusively of \(\mu \in \mathbb{Z}^+\) copies of \(S_n(0)\) where \(\mu\) depends on \(n\), \(k\) and the type of adjacency. We derive an expression for \(\mu\) for all types of adjacency.

The edge-distinguishing chromatic number \(\lambda(G)\) of a simple graph \(G\) is the minimum number of colors \(k\) assigned to the vertices in \(V(G)\) such that each edge \(\{u_i, u_j\}\) corresponds to a different set \(\{c(u_i), c(u_j)\}\). Al-Wahabi et al.\ derived an exact formula for the edge-distinguishing chromatic number of a path and of a cycle. We derive an exact formula for the edge-distinguishing chromatic number of a spider graph with three legs and of a spider graph with \(\Delta\) legs whose lengths are between 2 and \(\frac{\Delta+3}{2}\).

Motivated by the existing 3-equitable labeling of graphs, in this paper we introduce a new graph labeling called 3-equitable total labeling and we investigate the 3-equitable total labeling of several families of graphs such as \(C_n\), \(W_n\), \(SL_n\), \(S(K_{4,n})\) and \(K_n^{(4)}\).