Ioan Tomescu

Metric bases in digital geometry

Abstract Let S be a metric space under the distance function d. A metric basis is a subset B ⊆ S such that d(b, x) = d(b, y) for all b ϵ B implies x = y. It is shown that for Euclidean distance, the minimal metric bases for the digital plane are just the sets of three noncollinear points; but for city block or chessboard distance, the digital plane has no finite metric basis. The sizes of minimal metric bases for upright digital rectangles are also derived, and it is shown that there exist rectangles having minimal metric bases of any size ≥ 3.

Semi extremal properties of the degree distance of a graph

Abstract This paper deals with two conjectures made by Dobrynin and Kochetova on the minimum and maximum values of the degree distance of a graph: one of them is proved (by showing that K 1, n −1 is the unique extremal graph) and the other one is disproved.

Hypertrees and Bonferroni inequalities

In this paper some basic properties and examples of h-hypertrees are presented. It is also shown that this combinatorial structure is a natural tool for obtaining Bonferroni type inequalities which are equalities for some families of sets.

Chromatic Coefficients of Linear Uniform Hypergraphs

In this paper, formulae are given for the coefficients of the highest powers of?in the chromatic polynomialP(H, ?) of a linear uniform hypergraphH, thus generalizing the corresponding result of Meredith for graphs. The new result implies, among other things, that the elementaryh-uniform cycleChmwithmedges is chromatically unique for allm,h?3.

Maximal chromatic polynomials of connected planar graphs

In this paper we obtain chromatic polynomials of connected 3- and 4-chromatic planar graphs that are maximal for positive integer-valued arguments. We also characterize the class of connected 3-chromatic graphs having the maximum number of p-colorings for p ≥ 3, thus extending a previous result by the author (the case p = 3).

Digital metrics: A graph-theoretical approach

Consider the following two graphs M and N, both with vertex set Z x Z, where Z is the set of all integers. In M, two vertices are adjacent when their euclidean distance is 1, while in N, adjacency is obtained when the distance is either 1 or @/2. By definition, H is a metric subgraph of the graph G if the distance between any two points of H is the same as their distance in G. We determine all the metric subgraphs of M and N. The graph-theoretical distances in M and N are equal respectively to the city block and chessboard matrics used in pattern recognition.

Note: The Ramsey numbers of large cycles versus wheels

In this paper we show that the Ramsey number R(Cn ,W m) = 2n − 1 for even m and n 5m/2 − 1.

Path generated digital metrics

A new class of metrics, called path-generated metrics, is defined for the digital plane. There are essentially five different metrics in the class, including the classical city block and chessboard metrics.

Alternating 6-cycles in perfect matchings of graphs representing condensed benzenoid hydrocarbons

Abstract In this paper recurrence relations and algebraic expressions are deduced for the number of perfect matchings (Kekule structures) and of alternating 6-cycles for all perfect matchings of graphs composed from k linearly condensed portions consisting each of j +1 hexagons. These numbers are also expressed as polynomials in j , whose coefficients are rational polynomials in k which are found in an explicit form. An asymptotic ratio is obtained between the number of alternating 6-cycles in all perfect matchings and the total number of 6-cycles, as a function (40) of j . Some applications of these results to chemistry are presented, e.g. the ‘conjugated circuits method’ which gives resonance energies of condensed benzenoid hydrocarbons, and which depends mainly on the number of perfect matchings.

Maximum chromatic polynomials of 2-connected graphs

In this paper we obtain chromatic polynomials P(G; λ) of 2-connected graphs of order n that are maximum for positive integer-valued arguments λ ≧ 3. The extremal graphs are cycles Cn and these graphs are unique for every λ ≧ 3 and n ≠ 5. We also determine max{P(G; λ): G is 2-connected of order n and G ≠ Cn} and all extremal graphs relative to this property, with some consequences on the maximum number of 3-colorings in the class of 2-connected graphs of order n having X(G) = 2 and X(G) = 3, respectively. For every n ≧ 5 and λ ≧ 4, the first three maximum chromatic polynomials of 2-connected graphs are determined.