Minors in lifts of graphs

We study here lifts and random lifts of graphs, as defined by Amit and Linial (Combinatorica 22 (2002), 1–18). We consider the Hadwiger number η and the Hajós number σ of ?‐lifts of Kⁿ and analyze their extremal as well as their typical values (that is, for random lifts). When ? = 2, we show that , and random lifts achieve the lower bound (as n → ∞). For bigger values of ?, we show . We do not know how tight these bounds are, and in fact, the most interesting question that remains open is whether it is possible for η to be o(n). When ? < O(log n), almost every ?‐lift of Kⁿ satisfies η = Θ(n) and for , almost surely . For bigger values of ?, almost always. The Hajós number satisfies , and random lifts achieve the lower bound for bounded ? and approach the upper bound when ? grows. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Hamilton cycles in random lifts of graphs

An n-lift of a graph K is a graph with vertex set V(K)×[n], and for each edge (i,j)E(K) there is a perfect matching between {i}×[n] and {j}×[n]. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that there are constants h₁,h₂ such that if h≥h₁ then a random n-lift of the complete graph K_h is hamiltonian and if h≥h₂ then a random n-lift of the complete bipartite graph K_h,h is hamiltonian .     

Spectra and eigenspaces of arbitrary lifts of graphs

Journal of Algebraic Combinatorics - We describe, in a very explicit way, a method for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular...     

Random lifts of graphs: Independence and chromatic number

For a graph G, a random n‐lift of G has the vertex set V(G)×[n] and for each edge [u, v] ∈ E(G), there is a random matching between {u}×[n] and {v}×[n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G fixed and n→∞. For the independence number, upper and lower bounds are obtained as solutions to certain optimization problems on the base graph. For a base graph G with chromatic number χ and fractional chromatic number χ_f, we show that the chromatic number of typical lifts is bounded from below by const ? and also by const ? χ_f/log²χ_f (trivially, it is bounded by χ from above). We have examples of graphs where the chromatic number of the lift equals χ almost surely, and others where it is a.s. O(χ/logχ). Many interesting problems remain open. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 1–22, 2002     

Constructing cospectral graphs

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastn ³ g _–1 for some>0, whereg _n is the number of graphs of ordern andn 7.     

Constructing and classifying neighborhood anti-Sperner graphs

For a simple graph G let N_G(u) be the (open) neighborhood of vertex u∈V(G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v∈V(G)?{u} with N_G(u)⊆N_G(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if N_H(u)≠N_H(v) when u≠v, for all u and v∈V(H).In Porter and Yucas [T.D. Porter, J.L. Yucas. Graphs whose vertex-neighborhoods are anti-sperner, Bulletin of the Institute of Combinatorics and its Applications 44 (2005) 69-77] a characterization of regular NAS graphs was given: ‘each regular NAS graph can be obtained from a host graph by replacing vertices by null graphs of appropriate sizes, and then joining these null graphs in a prescribed manner’. We extend this characterization to all NAS graphs, and give algorithms to construct all NAS graphs from host ND graphs. Then we find and classify all connected r-regular NAS graphs for r=0,1,…,6.     

Constructing disjoint paths on expander graphs

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed.We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks.We present a sufficient condition for the existence ofKn ^Q edge-disjoint paths connecting any set ofK pairs of vertices on an expander graph, wheren is the number of vertices and<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C.Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which then participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them.Finally, we show how to apply variants of our parallel algorithms to find sets ofvertex-disjoint paths when certain conditions are satisfied.Supported in part by a Weizmann fellowship and by contract ONR N00014-85-C-0731.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

Constructing graphs with pairs of pseudo-similar vertices

Vertices u and v in the graph G are said to be pseudo-similar if G ? u ? G ? v but no automorphism of G maps u onto v. It is shown that a known procedure for constructing finite graphs with pairs of pseudo-similar vertices actually produces all such graphs. An additional procedure for constructing infinite graphs with pseudo-similar vertices is introduced and it is shown that all such graphs can be obtained by using either this or the first-mentioned method. The corresponding result for pseudo-similar edges is given.     

Constructing connected bicritical graphs with edge-connectivity 2

A graph $G$ is said to be bicritical if the removal of any pair of vertices decreases the domination number of $G$. For a bicritical graph $G$ with the domination number $t$, we say that $G$ is $t$-bicritical. Let $\lambda \left(G\right)$ denote the edge-connectivity of $G$. In [2], Brigham et al. (2005) posed the following question: If $G$ is a connected bicritical graph, is it true that $\lambda \left(G\right)\ge 3?$In this paper, we give a negative answer toward this question; namely, we give a construction of infinitely many connected $t$-bicritical graphs with edge-connectivity $2$ for every integer $t\ge 5$. Furthermore, we give some sufficient conditions for a connected $5$-bicritical graph to have $\lambda \left(G\right)\ge 3$.     

Constructing universal graphs for induced-hereditary graph properties

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m’th position of its binary expansion. It is well known that R is a universal graph in the set I of all countable graphs (since every graph in I is isomorphic to an induced subgraph of R). In this paper we describe a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of a universal graph for the set of finite graphs in a product of properties of graphs is also presented. The paper is concluded by a comparison between the two types of construction of universal graphs.     

Constructing few-weight linear codes and strongly regular graphs

Linear codes with few weights have applications in data storage systems, secret sharing schemes, graph theory and so on. In this paper, we construct a class of few-weight linear codes by choosing defining sets from cyclotomic classes and we also establish few-weight linear codes by employing weakly regular bent functions. Notably, we get some codes that are minimal and we also obtain a class of two-weight optimal punctured codes with respect to the Griesmer bound. Finally, we get a class of strongly regular graphs with new parameters by using the obtained two-weight linear codes.     

Constructing even radius tightly attached half-arc-transitive graphs of valency four

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41–76, 1998) classified odd radius tightly attached tetravalent half-arc-transitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K _4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.      

Constructing cospectral graphs via a new form of graph product

We present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.     

Constructing trees in graphs with no K2,s

Let s ≥ 2 be an integer and k > 12(s ? 1) an integer. We give a necessary and sufficient condition for a graph G containing no K_2,s with and to contain every tree T of order k + 1. We then show that every graph G with no K_2,s and average degree greater than k ? 1 satisfies this condition, improving a result of Haxell, and verifying a special case of the Erd?s—Sós conjecture, which states that every graph of average degree greater than k ? 1 contains every tree of order k + 1. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 301–310, 2007