The theory of lifting voltage digraphs provides a useful tool for constructing large digraphs with specified properties from suitable small base digraphs endowed with an assignment of voltages (= elements of a finite group) on arcs.
We revisit the degree/diameter problem for digraphs from this new perspective and prove a general upper bound on the diameter of a lifted digraph in terms of properties of the base digraph and voltage assignment.
In addition, we demonstrate that all currently known largest vertex-transitive Cayley digraphs for semidirect products of groups can be described by means of a voltage assignment construction using simpler groups.

The “characteristic” of a graph—the number of vertices, minus the number of edges, plus the number of triangles, etc.—is a little-studied, overtly combinatorial graph parameter intrinsically related to chordal graphs and common neighborhoods of subgraphs. I also introduce a sequence of related “higher characteristic” parameters.

A \({least \;deviant\; path}\) between two vertices in a weighted graph is defined as a path that minimizes the difference between the largest and smallest edge weights on the path.Algorithms are presented to determine the least deviant path. The fastest algorithm runs in \(O(|E|^{1.793})\), in the worst case. A type of two-dimensional binary search is used to achieve this running time.

An SOLS (self-orthogonal Latin square) of order \(n\) with \(n_i\) missing sub-SOLS (holes) of order \(h_i\) (\(1 \leq i \leq k\)), which are disjoint and spanning (i.e., \(\sum_{i=1}^{k} n_ih_i = n\)), is called a frame SOLS and denoted by \(\text{FSOLS}(h_1^{n_1}, h_2^{n_2}, \ldots, h_k^{n_k})\).In this article, it is shown that an \(\text{FSOLS}(3^{n-u}3^1)\) exists if and only if \(n \geq 4\) and \(n \geq 1 + \frac{2u}{3}\), with seventeen possible exceptions \((n, u) =\{(5, 1)\}\) and \(\{(n, u) = (n, \lfloor \frac{3(n-1)}{2}\rfloor)\) for \((n \in \{6, 10, 14, 18, 22, 30, 34, 38, 42, 46, 54, 58, 62, 66, 70, 94\} \).

It is straightforward to show that the full automorphism group of \(G \otimes K_n\) contains the Cartesian cross product of \(\text{Aut}(G)\) and \(S_n\). If \(\text{Aut}(G \otimes K_n)\) properly contains this cross product, then we will say that \(G \otimes K_n\) has a “rich” automorphism group. First, several conditions on \(G\) that ensure that \(G \otimes K_n\) has a rich automorphism group are given. Then, it is shown that these conditions are both necessary and sufficient for \(G \otimes K_n\) to have a rich automorphism group.

In this note, necessary and sufficient conditions are given for the existence of an equitable partial Steiner triple system \((S,T)\) on \(n\) symbols with exactly \(t\) triples, such that the leave of \((S,T)\) contains a \(1\)-factor if \(n\) is even and a near \(1\)-factor if \(n\) is odd.

A catalogue is presented which contains the graphs having order at most 10 which are critical with respect to the total chromatic number. A number of structural properties which cause these graphs to be critical are discussed, and a number of infinite classes of critical graphs are identified.A total colouring of a graph \(G\) is a function assigning colours to the vertices and edges of \(G\) in such a way that no two adjacent or incident elements are assigned the same colour. The total chromatic number, \(\chi”(G)\), is the minimum number of colours which need to be assigned to obtain a total colouring of the graph \(G\).
A longstanding conjecture, made independently by Behzad [3] and Vizing [17], claims that
\[
\Delta(G) + 1 \leq \chi”(G) \leq \Delta(G) + 2
\]
where \(\Delta(G)\) is the maximum degree of \(G\). The lower bound is sharp, the upper bound remains to be proved. A graph \(G\) is said to be Type 1 if \(\chi”(G) = \Delta(G) + 1\) and is said to be Type 2 if \(\chi”(G) \geq \Delta(G) + 2\).

We define a graph \(G\) to be critical with respect to the total chromatic number if \(G\) is connected and \(\chi”(G – e) < \chi''(G)\) for every edge \(e\) in \(G\). In Section 1 of this paper we identify all small order critical graphs, the catalogue of graphs is presented as a table of diagrams. In Section 2 we study structural properties of these graphs in order to identify features which cause a graph to be Type 2.

Given \(m\) unit-capacity bins and a collection \(x(n)\) of \(n\) pieces, each with a positive size at most one, the dual bin packing problem asks for packing a maximum number of pieces into the \(m\) bins so that no bin capacity is
exceeded. Motivated by the NP-hardness of the problem, Coffman et al. proposed a class of heuristics, the \({prefix}\) algorithms, and analyzed its worst-case performance bound.Bruno and Downey gave a probabilistic bound for the FFI algorithm (which is a prefix algorithm proposed by Coffman et al.), under the assumption that piece sizes are drawn from the uniform distribution over \([0, 1]\). In this article, we generalize their result: Let \(F\) be an \({arbitrary}\) distribution over \([0, 1]\), and let
\(x(n)\) denote a random sample of a random variable \(X\) distributed according to \(F\). Then, for any \(\varepsilon > 0\), there are \(\lambda > 0\) and \(N > 0\),
dependent only on \(m\), \(\varepsilon\), and \(F\), such that for all \(n \geq N\),
\begin{align*}
\Pr\left(\frac{{\mathrm{OPT}}(x(n), m)}{{\mathrm{PRE}}(x(n), m)} \leq 1 + \varepsilon\right)
&> 1 – Me^{-2\lambda n},
\end{align*}
where \(M\) is a universal constant.
Another probabilistic bound is also given for \(\frac{\mathrm{OPT}(x(n),m)}{\mathrm{PRE}(x(n),m)}\), under a
mild assumption of \(F\).

The distance of a vertex \(u\) in a connected graph \(G\) is defined by \(\sigma_G(u) := \sum_{v \in V(G)} d(u, v)\), and the distance of \(G\) is given by \(\sigma(G) = \frac{1}{2} \sum_{u \in V(G)} \sigma(u) (= \sum_{\{u,v\} \subseteq V(G)} d(u, v)\). Thus, the average distance between vertices in a connected graph \(G\) of order \(n\) is \(\frac{\sigma(G)}{\binom{n}{2}}\). These graph invariants have been studied for the past fifty years. Here, we discuss some known properties and present a few new results, together with several open problems. We focus on trees.

Let \(\chi^*(G)\) denote the minimum number of colors required in a coloring \(c\) of the vertices of \(G\) where for adjacent vertices \(u, v\) we have \(c(N_G[u]) \neq c(N_G[v])\) when \(N_G[u] \neq N_G[v]\) and \(c(u) \neq c(v)\) when \(N_G[u] = N_G[v]\). We show that the problem of deciding whether \(\chi^*(G) \leq n\), where \(n \geq 3\), is NP-complete for arbitrary graphs. We find \(\chi^*(G)\) for several classes of graphs, including bipartite graphs, complete multipartite graphs, as well as cycles and their complements. A sharp lower bound is given for \(\chi^*(G)\) in terms of \(\chi(G)\) and an upper bound is given for \(\chi^*(G)\) in terms of \(\Delta(G)\). For regular graphs with girth at least four, we give substantially better upper bounds for \(\chi^*(G)\) using random colorings of the vertices.