Denis Hanson

Vertex domination-critical graphs

A graph G is vertex domination-critical if for any vertex v of G the domination number of G - v is less than the domination number of G. If such a graph G has domination number γ, it is called γ-critical. Brigham et al. studied γ-critical graphs and posed the following questions: (1) If G is a γ-critical graph, is |V| ≥ (δ + 1)(γ - 1) + 1?(2) If a γ-critical graph G has (Δ + 1)(γ - 1) + 1 vertices, is G regular? (3) Does i = γ for all γ-critical graphs? (4) Let d be the diameter of the γ-critical graph G. Does d ≤ 2(γ - 1) always hold? We show that the first and third questions have a negative answer and the others have a positive answer.

Lower bounds on edge searching

Searching a network for intruders is an interesting and difficult problem. Edge-searching is one such search model, in which intruders may exist anywhere along an edge. Since finding the minimum number of searchers necessary to search a graph is NP-complete, it is natural to look for bounds on the search number. We show lower bounds on the search number using minimum degree, girth, chromatic number, and colouring number.

Star forests, dominating sets and Ramsey-type problems

A star forest of a graph G is a spanning subgraph of G in which each component is a star. The minimum number of edges required to guarantee that an arbitrary graph, or a bipartite graph, has a star forest of size n is determined. Sharp lower bounds on the size of a largest star forest are also determined. For bipartite graphs, these are used to obtain an upper bound on the domination number in terms of the number of vertices and edges in the graph, which is an improvement on a bound of Vizing. In turn, the results on bipartite graphs are used to determine the minimum number of lattice points required so that there exists a subset of n lattice points, no three of which form a right triangle with legs parallel to the coordinate axes.

On cages with given degree sets

Abstract We consider the problem of constructing minimal graphs of given girth having a particular degree set.

Edge‐colored saturated graphs

A graph G is (k1, k2, …, kt)-saturated if there exists a coloring C of the edges of G in t colors 1, 2, …, t in such a way that there is no monochromatic complete ki-subgraph K of color i, 1 ⩽ i ⩽ t, but the addition of any new edge of color i, joining two nonadjacent vertices in G, with C, creates a monochromatic K of color i, 1 ⩽ i ⩽ t. We determine the maximum and minimum number of edges in such graphs and characterize the unique extremal graphs.

Sum-free sets and Ramsey numbers

In this note we obtain new lower bounds for the Ramsey numbers R(5, 5) and R(5, 6). The method is based on computational results of partitioning the integers into sum-free sets. We obtain R(5, 5)@?42 and R(5, 6)@?53.

On finite Δ-systems II

Abstract A Δ(k) system is a family F of k distinct sets which have pairwise the same intersection. A weak Δ(k) system is a family F of k distinct sets such that | F þ G | = t for some non-negative integer t and all F,G ϵ F , F ≠ G. In this paper we study some functions related to these Δ-systems. In particular α(n,k) = max{| F | : | F | = n ∀ F ϵ F , F ⊅ weak Δ(k)} then α(3,3) = 10, α(m + n, k) ⩾ α (m, k) α(n, k) and α (n, 3) > c5 n 2

Time constrained graph searching

We consider time constraints for four models of searching graphs for intruders. One model is the standard cops and robber vertex-searching model with complete visibility. The second model differs from the preceding one only in that none of the searchers can see the intruder. The third model is a vertex-searching model in which searchers and an intruder move simultaneously and none of the searchers can see the intruder. The fourth model is simultaneous edge searching with an arbitrarily fast intruder.

The size of a minimum five-chromatic K 4 -free graph

The minimum number of vertices in a five-chromatic graph that does not contain K4 is determined. Nenov [3] has shown that the chromatic number of any ten-vertex graph without K, is at most four. A well-known construction of Mycielski (see [l]) leads to the Griitzsch graph, an eleven-vertex, four-chromatic, triangle-free graph. Adding a new vertex adjacent to all the vertices of the Griitzsch graph creates a twelve-vertex, five-chromatic graph, not containing K,. Thus the minimum number of vertices in a five-chromatic, K,-free graph is either eleven or twelve. We show that the correct answer is eleven by describing two examples. Toft drew attention to this problem in his book Graph Co/owing Problems, [4], Chapter 6. In a later paper [4], he again referred to the problem, but now as solved. Two simultaneous independent solutions are mentioned; one by Youngs [6], and one by Jensen and Royle [3]. The latter example was also found, independently, by Hanson and MacGillivray. Both of these examples are reported here. According to Jensen and Royle [3], who used a computer search, there are 56 non-isomorphic, five-chromatic, K,-free graphs on eleven vertices. The example found by Youngs is the graph G shown in Fig. 1. One may check directly that G is K,-free. (In fact, it is the unique graph on eleven vertices that is both five-critical and four-saturated, that is, the addition of any edge creates a copy of K,,

A note on minimum graphs with girth pair (4, 2/+1)

We show that the size of a smallest connected k-regular graph with girth pair (4, 2l + 1) is within a constant of (2l + 1) k/2. In so doing we disprove a conjecture of Harary and Kovacs.