Aleksander Vesel

Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2,1)-colorings and independence numbers

Rotagraphs generalize all standard products of graphs in which one factor is a cycle. A computer-based approach for searching graph invariants on rotagraphs is proposed and two of its applications are presented. First, the λ-numbers of the Cartesian product of a cycle and a path are computed, where the λ-number of a graph G is the minimum number of colors needed in a (2, 1)-coloring of G. The independence numbers of the family of the strong product graphs C7 × C7 × C2k+1 are also obtained.

L(2,1)-labeling of strong products of cycles

In a wireless network, different devices are fitted with radio transmitters and receivers. The task of channel assignment problem is to assign radio frequencies to transmitters at different locations, without interference [4]. The problem is closely related to graph labeling where the vertices of a graph represent the transmitters and the adjacencies show possible interference. However, the interference phenomena may be so powerful that even the different channels used at close stations may interfere, if the channels are too near. Thus, in many cases, channels assigned to stations must be separated according to the distance between two transmitters. L(2,1)-labeling, a variation of the channel assignment problem, distinguishes between close transmitters which must receive different

L(2,1)-labeling of direct product of paths and cycles

An L (2, 1)-labeling of a graph G is an assignment of labels from {0, 1,....., λ} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The λ-number λ(G) of G is the minimum value λ such that G admits an L (2, 1)-labeling. Let G × H denote the direct product of G and H. We compute the λ-numbers for each of C7i × C7j, C11i × C11j × C11k, P4 × Cm, and P5 × Cm. We also show that for n ≥ 6 and m ≥ 7, λ(Pn × Cm) = 6 if and only if m = 7k, k ≥ 1. The results are partially obtained by a computer search.

Fast Recognition of Fibonacci Cubes

Abstract Fibonacci cubes are induced subgraphs of hypercubes based on Fibonacci strings. They were introduced to represent interconnection networks as an alternative to the hypercube networks. We derive a characterization of Fibonacci cubes founded on the concept of resonance graphs. The characterization is the basis for an algorithm which recognizes these graphs in O(mlog n) time.

Optimal L ( d ,1)-labelings of certain direct products of cycles and Cartesian products of cycles

An L(d,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices such that adjacent vertices receive labels that differ by at least d and those at a distance of two receive labels that differ by at least one, where d ≥ 1. Let λ1d(G) denote the least λ such that G admits an L(d,1)-labeling using labels from {0, 1, ..., λ}. We prove that (i) if d ≥ 1, k ≥ 2 and m0, ..., mk-1 are each a multiple of 2k + 2d - 1, then λ1d(Cm0 × ... × Cmk-1) ≤ 2k + 2d - 2, with equality if 1 ≤ d ≤ 2k, and (ii) if d ≥ 1, k ≥ 1 and m0, ..., mk-1 are each a multiple of 2k + 2d - 1, then λ1d (Cm0□ ...□Cmk-1) ≤ 2k + 2d - 2, with equality if 1 ≤ d ≤ 2k.

On the packing chromatic number of square and hexagonal lattice

The packing chromatic number χ ρ ( G ) of a graph G is the smallest integer k such that the vertex set V ( G ) can be partitioned into disjoint classes X 1 , …,  X k , with the condition that vertices in X i have pairwise distance greater than i . We show that the packing chromatic number for the hexagonal lattice ℋ is 7. We also investigate the packing chromatic number for infinite subgraphs of the square lattice ℤ  2 with up to 13 rows. In particular, we establish the packing chromatic number for P 6 □ ℤ and provide new upper bounds on these numbers for the other subgraphs of interest. Finally, we explore the packing chromatic number for some infinite subgraphs of ℤ  2 □  P 2 . The results are partially obtained by a computer search.

Improved lower bound on the Shannon capacity of C 7

An independent set with 108 vertices in the strong product of four 7-cycles (C7⊠C7⊠C7⊠C7) is given. This improves the best known lower bound for the Shannon capacity of the graph C7 which is the zero-error capacity of the corresponding noisy channel. The search was done by a computer program using the “simulated annealing” algorithm with a constant time temperature schedule.

Binary coding of Kekulé structures of catacondensed benzenoid hydrocarbons.

An algorithm is described by means of which the Kekulé structures of a catacondensed benzenoid molecule (with h hexagons) are transformed into binary codes (of length h). By this, computer-aided manipulations with, and memory-storage of Kekulé structures are much facilitated. Any Kekulé structure can easily be recovered from its binary code.

Linear Recognition and Embedding of Fibonacci Cubes

Fibonacci strings are binary strings that contain no two consecutive 1s. The Fibonacci cube Γh is the subgraph of the h-cube induced by the Fibonacci strings. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. We derive a new characterization of Fibonacci cubes. The characterization is the basis for an algorithm which recognizes these graphs in linear time. Moreover, a graph which was recognized as a Fibonacci cube can be embedded into a hypercube using Fibonacci strings within the same time bound.

The independence number of the strong product of odd cycles

Abstract The independence number of the strong product C 5 ⊠C 7 ⊠C 7 determined by the NISPOC software package is presented. Better lower bounds on the independence numbers for two infinite families of strong products of three odd cycles are given.