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.

A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \(\Theta(n^2)\) algorithm which computes all the weak repetitions in a given string of length \(n\) defined
on an arbitrary alphabet \(A\). Using results on Fibonacci and other simple strings, we prove that this algorithm is asymptotically optimal over all known encodings of the output.

An algorithm is presented which, when given the non-isomorphic designs with given parameters, generates all the trades in each of the designs. The lists of trades generated by the algorithm were used to find the sizes, previously unknown, of smallest defining sets of the \(21\) non-isomorphic \(2\)-(10, 5, 4) designs. Consideration of trades in a design to isomorphic and to non-isomorphic designs led to two variations on the concept of defining sets. The lists of trades were then used to find the sizes of these smallest member and class defining sets, for five parameter sets.