For an integer $\ell \ge 2$, the $\ell$-connectivity ($\ell$-edge-connectivity) of a graph $G$ of order $p$ is the minimum number of vertices (edges) that need to be deleted from $G$ to produce a disconnected graph with at least $\ell$ components or a graph with at most $\ell -1$ vertices. Let $G$ be a graph of order $p$ and $\ell$-connectivity ${\kappa }_{\ell }$. For each $k\in \{0,1,\dots ,{\kappa }_{\ell }\}$, let $s_k$ be the minimum $\ell$-edge-connectivity among all graphs obtained from $G$ by deleting $k$ vertices from $G$. Then $f_{\ell }=\{(0,s_0),\dots ,({\kappa }_{\ell },s_{{\kappa }_{\ell }})\}$ is the $\ell$-connectivity function of $G$. The $\ell$-connectivity functions of complete multipartite graphs and caterpillars are determined.

An infinite class of graphs is constructed to demonstrate that the difference between the independent domination number and the domination number of $3$-connected cubic graphs may be arbitrarily large.

The domination number $\gamma (G)$ and the total domination number ${\gamma }_t(G)$ of a graph are generalized to the $K_n$-domination number ${\gamma }_{k_n}(G)$ and the total $K_n$-domination number ${\gamma }_{K_n}^t(G)$ for $n\ge 2$, where $\gamma (G)={\gamma }_{K_2}(G)$ and ${\gamma }_t(G)={\gamma }_{K_2}^t(G)$. A nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers is said to be a $K_n$-domination (total $K_n$-domination, respectively) sequence if it can be realized as the sequence of generalized domination (total domination, respectively) numbers ${\gamma }_{K_2}(G),{\gamma }_{K_3}(G),\dots ,{\gamma }_{K_m}(G)$ (${\gamma }_{K_2}^t(G),{\gamma }_{K_3}^t(G),\dots ,{\gamma }_{K_m}^t(G)$, respectively) of some graph $G$. It is shown that every nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers is a $K_n$-domination sequence (total $K_n$-domination sequence, if $a_2\ge 2$, respectively). Further, the minimum order of a graph $G$ with $a_2,a_3,\dots ,a_m$ as a $K_n$-domination sequence is determined. $K_n$-connectivity is defined and for $a_2\ge 2$ we establish the existence of a $K_m$-connected graph with $a_2,a_3,\dots ,a_m$ as a $K_n$-domination sequence for every nondecreasing sequence $a_2,a_3,\dots ,a_m$ of positive integers.

We study the problem of scheduling parallel programs with conditional branching on parallel processors. The major problem in having conditional branching is the non-determinism since the direction of a branch may be unknown until the program is midway in execution. In this paper, we overcome the problem of non-determinism by proposing a probabilistic model that distinguishes between branch and precedence relations in parallel programs. We approach the problem of scheduling task graphs that contain branches by representing all possible execution instances of the program by a single deterministic task graph, called the representative task graph. The representative task graph can be scheduled using one of the scheduling techniques used for deterministic cases. We also show that a schedule for the representative task graph can be used to obtain schedules for all execution instances of the program.

We give a list of all graphs of maximum degree three and order at most sixteen which are critical with respect to the total chromatic number.

In this paper we give a partial answer to a query of Lindner conceming the quasigroups arising from $2$-perfect $6$-cycle systems.

Consider the paths ${\pi }_t(i_1),\dots ,{\pi }_t(i_k)$ from the root to the leaves $i_1,\dots ,i_k$ in a random binary tree $t$ with $n$ internal nodes, where all such trees are assumed equally likely and the leaves are enumerated from left to right. We investigate, for fixed $i_1,\dots ,i_k$ and $n$, the average size of ${\pi }_t(i_1)\cup \dots \cup {\pi }_t(i_k)$ resp. of ${\pi }_t(i_1)\cap \dots \cap {\pi }_t(i_k)$ (the latter corresponding to the average depth of the smallest subtree containing $i_1,\dots ,i_k$). By a rotation argument, both problems are reduced to the case $k=1$, for which a solution is known. Furthermore, formulas for the probability distributions of the depth of leaf $i$, the distance between leaf $i$ and $j$ and the length of ${\pi }_t(i)\cap {\pi }_t(j)$ are derived.

Chetwynd and Hilton made the following $edge-colouringconjecture$: if a simple graph $G$ satisfies $\mathrm{\Delta }(G)>\frac{1}{3}|V(G)|$, then $G$ is Class $2$ if and only if it contains an overfull subgraph $H$ with $\mathrm{\Delta }(H)=\mathrm{\Delta }(G)$. They also made the following $total-colouringconjecture$: if a simple graph $G$ satisfies $\mathrm{\Delta }(G)\ge \frac{1}{2}(|V(G)|+1)$, then $G$ is Type $2$ if and only if $G$ contains a non-conformable subgraph $H$ with $\mathrm{\Delta }(H)=\mathrm{\Delta }(G)$. Here we show that if the edge-colouring conjecture is true for graphs of even order satisfying $\mathrm{\Delta }(G)>\frac{1}{2}|V(G)|$, then the total-colouring conjecture is true for graphs of odd order satisfying $\delta (G)\ge \frac{3}{4}|V(G)|–\frac{1}{4}$ and $\text{def}(G)\ge 2(\mathrm{\Delta }(G)–\delta (G)+1)$.