Peter J. Owens

Shortness parameters of families of regular planar graphs with two or three types of face

This paper is concerned with non-Hamiltonian planar graphs. It is shown that the class of 3-connected cubic planar graphs whose faces are all pentagons or 10-gons contains non-Hamiltonian members and that the shortness coefficient of this class of graphs is less than unity. For several classes of 3-connected regular planar graphs, of valency 4 or 5, it is shown that the shortness exponent is less than unity.

Simple 3-polytopal graphs with edges of only two types and shortness coefficients

Abstract We consider classes of simple 3-polytopal graphs whose edges are incident with either two 5-gons or a 5-gon and a q -gon ( q > 5). We show that the shortness coefficient is less than one for all q ⩾ 28, settle a question raised by Jendroľ and Tkac in a recent paper in the journal and prove that all the graphs in these classes are cyclically 4-edge-connected.

On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces

Abstract. We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path Pk, a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m.

Longest cycles in generalized Buckminsterfullerene graphs

It is shown that every regular 3-valent polyhedral graph whose faces are all 5-gons and 6-gons contains a cycle through at least 4/5 of its vertices.

Non-hamiltonian 5 4 -tough maximal planar graphs

Abstract By a theorem of Whitney (1931) the toughness t ( G ) of a non-hamiltonian maximal planar graph G is less than or equal to 3 2 . Improving a result of Dillencourt (1991), it is shown that the shortness exponent of the class of maximal planar graphs with toughness greater than or equal to 5 4 is less than 1.

SHORTNESS PARAMETERS FOR POLYHEDRAL GRAPHS

Abstract We survey results about shortness parameters for families of polyhedral graphs and mention some open problems and conjectures.

Aspects of complete sets of 9 × 9 pairwise orthogonal latin squares

Abstract An affine plane of order 9 can be specified by an orthogonal array with 10 constraints and 9 levels. A complete set of pairwise orthogonal 9 × 9 latin squares is obtained when any two of the constraints are taken as rows and columns. Any 3 of the 10 constraints give rise to an adjugacy set of 9 × 9 latin squares from a particular species. For each of the 7 affine planes of order 9 we count the occurrences of different species amongst the 120 subsets of 3 constraints. We give some properties of these species, including the orders of their automorphism groups. We verify the numbers of subplanes of order 2 in each of the 4 projective planes of order 9.

5-regular 3-polytopal graphs with edges of only two types and shortness exponents less than one

Abstract It is shown that, if q ⩾ 29 and q ≢ 0 (mod 3), the infinite class of 5-regular 3-polytopal graphs whose edges are incident with either two triangles or a triangle and a q -gon contains nonhamiltonian members and even has shortness exponent less than one.

Pentagonal 3-polytopal graphs with edges of only two types and shortness parameters

We consider the class of pentagonal 3-polytopal graphs all of whose edges are incident either with two 3-valent vertices or with a 3-valent vertex and a q-valent vertex. For most values of q, (i) we find a small non-hamiltonian graph in the class and (ii) we show that the shortness exponent of the class and the shortness coefficient of a special subclass are less than one. For q=4, we find a positive lower bound for the shortness coefficient.

Knight's move squares

Abstract A strict knights move square is a latin square in which any two cells which contain the same symbol are joined by a sequence of knights moves using only cells containing that symbol. We determine the values of n for which there exists a strict knights move square of order n .