Two problems on independent sets in graphs

Let i_t(G) denote the number of independent sets of size t in a graph G. Levit and Mandrescu have conjectured that for all bipartite G the sequence (i_t(G))_t≥0 (the independent set sequence of G) is unimodal. We provide evidence for this conjecture by showing that this is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph G(n,n,p), and show that for any fixed p∈(0,1] its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for .We also consider the problem of estimating i(G)=∑_t≥0i_t(G) for G in various families. We give a sharp upper bound on the number of independent sets in an n-vertex graph with minimum degree δ, for all fixed δ and sufficiently large n. Specifically, we show that the maximum is achieved uniquely by K_δ,n−δ, the complete bipartite graph with δ vertices in one partition class and n−δ in the other.We also present a weighted generalization: for all fixed x>0 and δ>0, as long as n=n(x,δ) is large enough, if G is a graph on n vertices with minimum degree δ then ∑_t≥0i_t(G)x^t≤∑_t≥0i_t(K_δ,n−δ)x^t with equality if and only if G=K_δ,n−δ.     

The diameter of sparse random graphs

We derive an expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence. The result holds asymptotically almost surely, assuming that certain convergence and supercriticality conditions are met, and is applicable to the classical random graph G_n,p with np = Θ(1) + 1, as well as certain random power law graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

A functional limit theorem for random graphs with applications to subgraph count statistics

We consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. the proof is based on a martingale convergence theorem. the evolving random graph allows us to study both the random graph model K_{n, p}, by fixing attention to a fixed time, and the model K_{n, N}, by studying it at the random time it contains exactly N edges. in particular, we obtain the asymptotic distribution as n → ∞ of the number of subgraphs isomorphic to a given graph G, both for K_{n, p} (p fixed) and K_{n, N} (N/(ⁿ₂)→ p). the results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers of n (the variance grows slower for K_n, N; the powers of n usually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. in some exceptional cases, the limit distribution is not normal.     

Folding Wheels and Fans

 If two non-adjacent vertices of a connected graph that have a common neighbor are identified and the resulting multiple edges are reduced to simple edges, then we obtain another graph of order one less than that of the original graph. This process can be repeated until the resulting graph is complete. We say that we have folded the graph onto complete graph. This process of folding a connected graph G onto a complete graph induces in a very natural way a partition of the vertex-set of G. We denote by F(G) the set of all complete graphs onto which G can be folded. We show here that if p and q are the largest and smallest orders, respectively, of the complete graph in F(W _n ) or F(F _n ), then K _s is in F(W _n ) or F(F _n ) for each s, q≤s≤p. Lastly, we shall also determine the exact values of p and q. Received: October, 2001 Final version received: June 26, 2002     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

Matching graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M₁ and M₂ of M(G) are adjacent if and only if |M₁ − M₂| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 73–86, 1998     

Hadwiger number and chromatic number for near regular degree sequences

We consider a problem related to Hadwiger's Conjecture. Let D=(d₁, d₂, …, d_n) be a graphic sequence with 0?d₁?d₂?···?d_n?n?1. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. Define h(D)=max{h(G): G∈R[D]} and χ(D)=max{χ(G): G∈R[D]}, where h(G) and χ(G) are Hadwiger number and chromatic number of a graph G, respectively. Hadwiger's Conjecture implies that h(D)?χ(D). In this paper, we establish the above inequality for near regular degree sequences. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 175–183, 2010     

Moments of graphs in monotone families

The kth moment of the degree sequence d₁ ≥ d₂ ≥ …d_n of a graph G is . We give asymptotically sharp bounds for μ_k(G) when G is in a monotone family. We use these results for the case k = 2 to improve a result of Pach, Spencer, and Tóth [15]. We answer a question of Erd?s [9] by determining the maximum variance of the degree sequence when G is a triangle‐free n‐vertex graph. © 2005 Wiley Periodicals, Inc.     

A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

We consider non-overlapping subgraphs of fixed order in the random graph K_n, p(n). Fix a strictly strongly balanced graph G. A subgraph of K_{n, p(n)} isomorphic to G is called a G-subgraph. Let X_n be the number of G-subgraphs of K_{n, p(n)} vertex disjoint to all other G-subgraphs. We show that if E[X_n]→∞ as n→, then X_n/E[X_n] converges to 1 in probability. Also, if E[X_n]→c as n→∞, then X_n satisfies a Poisson limit theorem. the Poisson limit theorem is shown using a correlation inequality similar to those appeared in Janson, ?uczak, and Ruciñski[8] and Boppana and Spencer [4].     

On the parity of crossing numbers

For an integer n ? 1, a graph G has an n-constant crossing number if, for any two good drawings ? and ?′ of G in the plane, μ(?) ≡ μ(?′) (mod n), where μ(?) is the number of crossings in ?. We prove that, except for trivial cases, a graph G has n-constant crossing number if and only if n = 2 and G is either K_p or K_q,r, where p, q, and r are odd.     

Equitable coloring of random graphs

An equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ₌(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ₌*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n^?1/5+? < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ₌(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n^?(1??) < p < 0.99 for some 0 < ?, then almost surely χ(G(n,p)) ≤ χ₌* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if < p < 0.99 for some 0 < ?, then almost surely we have χ(G(n,p)) ≤ χ₌*(G(n,p)) = O_?(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

1-Pancyclic Hamilton Cycles in Random Graphs

We consider a generalization of the pancyclic property. A graph G is defined to be 1-pancyclic if there is some Hamilton cycle H in G such that we can find a cycle C_s of length s (3 ? s ? n ? 1) using only the edges of H and one other edge e_s. We show that the threshold for G_n,p to be Hamiltonian, is the threshold for the 1-pancyclic property.     

Szemerédi's partition and quasirandomness

In this paper we shall investigate the connection between the Szemerédi Regularity Lemma and quasirandom graph sequences, defined by Chung, Graham, and Wilson, and also, slightly differently, by Thomason. We prove that a graph sequence (G_n) is quasirandom if and only if in the Szemerédi partitions of G_n almost all densities are ½ + o(l).     

The greedy coloring is a bad probabilistic algorithm

Given a graph G and an ordering p of its vertices, denote by A(G, p) the number of colors used by the greedy coloring algorithm when applied to G with vertices ordered by p. Let , , Δ be positive constants. It is proved that for each n there is a graph G_n such that the chromatic number of G_n is at most n, but the probability that A(G_n, p) < (1 − )n/log₂ n for a randomly chosen ordering p is O(n^−Δ).     

Idempotent algebras with log-linear free spectra

In this paper we consider idempotent algebras whosep _n -sequences (the sequences of the numbersp _n of essentiallyn-ary polynomials) have a subexponential rate of growth. Studying the symmetry groups of polynomials we establish some consequences of this property. In particular, a new characterization of semilattices is obtained, and all idempotent commutative algebras with log-linear free spectra are described.To Professor Jerzy Plonka in honour of his 60th birthdayPresented by Joel Berman.     

Partitioning a graph into convex sets

Let G be a finite simple graph. Let S⊆V(G), its closed interval I[S] is the set of all vertices lying on shortest paths between any pair of vertices of S. The set S is convex if I[S]=S. In this work we define the concept of a convex partition of graphs. If there exists a partition of V(G) into p convex sets we say that G is p-convex. We prove that it is NP-complete to decide whether a graph G is p-convex for a fixed integer p≥2. We show that every connected chordal graph is p-convex, for 1≤p≤n. We also establish conditions on n and k to decide if the k-th power of a cycle C_n is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.     

Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

A property of trees in terms of unique connected subgraphs

The following theorem is proved: Let G be a graph with p ≥ 3 points such that for some n, 3 ≤ n ≤ p, any n points lie on a unique smallest connected subgraph. Then G = C_n+1 or G is a tree, and conversely.     

The phase transition in the cluster‐scaled model of a random graph

For 0 < p < 1 and q > 0 let G_q(n,p) denote the random graph with vertex set [n]={1,…,n} such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that G_q(n,p)=G is proportional to . The first systematic study of G_q(n,p) was undertaken by 6 , who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of G_q(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The order of the largest complete minor in a random graph

Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G_n,p denote a random graph on n vertices with edge probability p. Bollobás, Catlin, and Erd?s (Eur J Combin 1 (1980), 195–199) asymptotically determined ccl(G_n,p) when p is a constant. ?uczak, Pittel and Wierman (Trans Am Math Soc 341 (1994) 721–748) gave bounds on ccl(G_n,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C such that whenever C/n < p < 1 ‐ ε then asymptotically almost surely ccl(G_n,p) = (1 ± ε)n/ , where b := 1/(1 ‐ p). If p = C/n for a constant C > 1, then ccl(G_n,p) = Θ( ). This extends the results in (Bollobás, Catlin, and P. Erd?s, Eur J Combin 1 (1980), 195–199) and answers a question of Krivelevich and Sudakov (preprint, 2006). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008