Sandi Klavar

On median nature and enumerative properties of Fibonacci-like cubes

Fibonacci cubes, extended Fibonacci cubes, and Lucas cubes are induced subgraphs of hypercubes defined in terms of Fibonacci strings. It is shown that all these graphs aremedian. Several enumeration results on the number of their edges and squares are obtained. Some identities involving Fibonacci and Lucas numbers are also presented.

Nonrepetitive colorings of trees

A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by @p(G). A famous theorem of Thue asserts that @p(P)=3 for any path P with at least four vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that @p(T) is bounded by 4 in this class we aim to describe the 4-chromatic trees. In particular, we study the 4-critical trees which are minimal with respect to this property. Though there are many trees T with @p(T)=4 we show that any of them has a sufficiently large subdivision H such that @p(H)=3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edge-colored by at most @D+1 colors without repetitions on paths.

On distance-balanced graphs

It is shown that the graphs for which the Szeged index equals @?G@?@?|G|^24 are precisely connected, bipartite, distance-balanced graphs. This enables us to disprove a conjecture proposed in [M.H. Khalifeh, H. Yousefi-Azari, A.R. Ashrafi, S.G. Wagner, Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149-1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of graph. Infinite families of distance-balanced, non-regular graphs that are prime with respect to the Cartesian product are also constructed.

On the geodetic number and related metric sets in Cartesian product graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products have product structure and that the contour set of a Cartesian product is geodetic if and only if their projections are geodetic sets in factors.

Wiener index versus Szeged index in networks

Let (G,w) be a network, that is, a graph G=(V(G),E(G)) together with the weight function w:E(G)->R^+. The Szeged index Sz(G,w) of the network (G,w) is introduced and proved that Sz(G,w)>=W(G,w) holds for any connected network where W(G,w) is the Wiener index of (G,w). Moreover, equality holds if and only if (G,w) is a block network in which w is constant on each of its blocks. Analogous result holds for vertex-weighted graphs as well.

Wiener index in weighted graphs via unification of Θ*-classes

It is proved that the Wiener index of a weighted graph (G,w) can be expressed as the sum of the Wiener indices of weighted quotient graphs with respect to an arbitrary combination of @Q^*-classes. Here @Q^* denotes the transitive closure of Djokovic-Winklers relation @Q. A related result for edge-weighted graphs is also given and a class of graphs studied in Yousefi-Azari et al. (2011) [25] is characterized as partial cubes.

Fast recognition algorithms for classes of partial cubes

Isometric subgraphs of hypercubes, or partial cubes as they are also called, are a rich class of graphs that include median graphs, subdivision graphs of complete graphs, and classes of graphs arising in mathematical chemistry and biology. In general, one can recognize whether a graph on n vertices and m edges is a partial cube in O(mn) steps, faster recognition algorithms are only known for median graphs. This paper exhibits classes of partial cubes that are not median graphs but can be recognized in O(mlogn) steps. On the way relevant decomposition theorems for partial cubes are derived, one of them correcting an error in a previous paper (Eur. J. Combin. 19 (1998) 677).

The index of a binary word

A binary word u is f-free if it does not contain f as a factor. A word f is d-good if for any f-free words u and v of length d, v can be obtained from u by complementing one by one the bits of u on which u and v differ, such that all intermediate words are f-free. We say that f is good if it is d-good for any d>=1. A word is bad if it is not good. The index @b(f) of f is the smallest integer d such that f is not d-good, so that @b(f)<~ if and only if f is bad. It is proved that @b(f)<|f|^2 holds for any bad word f. In addition, @b(f)<2|f| holds for almost all bad words f and it is conjectured that the same holds for all bad words. We construct an infinite family of 2-isometric bad words. It is conjectured that the words of this family are all the words that are bad and 2-isometric among those with exactly two 1s. These conjectures are supported by computer experiments.

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x@?V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.

On the role of hypercubes in the resonance graphs of benzenoid graphs

The resonance graph R(B) of a benzenoid graph B has the perfect matchings of B as vertices, two perfect matchings being adjacent if their symmetric difference forms the edge set of a hexagon of B. A family P of pair-wise disjoint hexagons of a benzenoid graph B is resonant in B if B-P contains at least one perfect matching, or if B-P is empty. It is proven that there exists a surjective map f from the set of hypercubes of R(B) onto the resonant sets of B such that a k-dimensional hypercube is mapped into a resonant set of cardinality k.