The third author proved earlier [8] that if a Euclidean space is colored with red and blue so that the distance one is forbidden for blue, and translates of some \(k\)-point configuration are forbidden for red, then the unit-distance chromatic number of the space is no greater than \(k\). Here we give a generalization.

We continue the study of graphs defined by a certain adjacency property by investigating the $n$-existentially closed line-critical graphs. We classify the \(1\)-e.c. line-critical graphs and give examples of \(2\)-e.c. line-critical graphs for all orders \(\geq 9\).

An isometric path is merely any shortest path between two vertices. Inspired by the game of `Cops and Robber’ and a result by Aigner \(\&\) Fromme [1], we are interested in determining the minimum number of isometric paths required to cover the vertices of a graph. We find a lower bound on this number in terms of the diameter of a graph and find the exact number for trees and grid graphs.

An edge-graceful \((p, q)\)-graph \(G = (V, E)\) is a graph with \(p\) vertices and \(q\) edges for which there is a bijection \(f : E \to \{1,2,\ldots,q\}\) such that the induced mapping \(f^+ : V \to \mathbb{Z}_p\), defined by \(f^+(u) \equiv \sum\limits_{uv \in E} f(uv) \pmod{p}\), for \(u \in V\), is a bijection. In this paper, some results on edge-gracefulness of trees are extended to \(k\)-fold graphs based on graphs with \(p$ vertices and \(p – 1\) edges. A \(k\)-fold multigraph \(G[k]\) derived from a graph \(G\) is one in which each edge of \(G\) has been replaced by \(k\) parallel edges with the same vertices as the original edge. Certain classes of \(k\)-fold multigraphs derived from paths, combs, and spiders are shown to be edge-graceful, as well as other graphs constructed by combining these graphs in specified ways.

We determine solutions to the problem of gossiping in minimum time (briefly: minimum time problem or MTP) which require less calls than the previously known solutions for infinitely many values of the number \(n\) of persons and optimal solutions to the MTP, i.e. solutions of the MTP which minimize the number of calls, for some values of \(n\). We conjecture that our methods provide optimal solutions of the MTP for all \(n\).

Erdős and Gallai (1963) showed that any \(r\)-regular graph of order \(n\), with \(r < n-1\), has chromatic number at most \({3n}/{5}\), and this bound is achieved by precisely those graphs with complement equal to a disjoint union of 5-cycles.

We are able to generalize this result by considering the problem of determining a \((j-1)\)-regular graph \(G\) of minimum order \(f(j)\) such that the chromatic number of the complement of \(G\) exceeds \({f(j)}/{2}\). Such a graph will be called an \(F(j)\)-\emph{graph}. We produce an \(F(j)\)-graph for all odd integers \(j \geq 3\) and show that \(f(j) = {5(j – 1)}/{2}$ if \(j \equiv 3 \pmod{4}\), and \(f(j) = 1 + {5(j – 1)}/{2}\) if \(j \equiv 1 \pmod{4}\).

A lemma of Enomoto, Llado, Nakamigawa and Ringel gives an upper bound for the edge number of a super edge-magic graph with \(p > 1\) vertices. In this paper we give some results which come out from answering some natural questions suggested by this useful lemma.

The scheme associated with a graph is an association scheme if and only if the graph is strongly regular. Consider the problem of extending such an association scheme to a superscheme in the case of a colored, directed graph. The obstacles can be expressed in terms of \(t\)-vertex conditions. If a graph does not satisfy the \(t\)-vertex condition, a prescheme associated with it cannot be erected beyond the \((t-3)\)rd-level.

A mandatory representation design MRD \((K; v)\) is a pairwise balanced design PBD \((K; v)\) in which for each \(k \in K\) there is at least one block in the design of size \(k\). The study of the mandatory representation designs is closely related to that of subdesigns in pairwise balanced designs. In this paper, we survey the known results on MRDs and pose some open questions.

It is shown that the necessary conditions are sufficient for the existence of all \(c\)-BRDs\((v, 3, \lambda)\) for negative \(c\)-values. This completes the study of \(c\)-BRDs with block size three as previously the authors and J. Seberry have shown that the necessary conditions are sufficient for \(c \geq -1\).