Distance-transitive dihedrants

The main result of this article is a classification of distance-transitive Cayley graphs on dihedral groups. We show that a Cayley graph X on a dihedral group is distance-transitive if and only if X is isomorphic to one of the following graphs: the complete graph K _2n ; a complete multipartite graph K _t×m with t anticliques of size m, where t m is even; the complete bipartite graph without 1-factor K _n,n − nK ₂; the cycle C _2n ; the incidence or the non-incidence graph of the projective geometry PG _d-1(d,q), d ≥ 2; the incidence or the non-incidence graph of a symmetric design on 11 vertices.     

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least cn^{1-(v_H-2)/(e_H-1)}(logn)^1/(e_H-1)cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}. This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs K _r,r with r≥5. When H is a complete graph K _s with s≥5 we show that for some C>0, with high probability the independence number of G is at most Cn^2/(s+1)(logn)^1-1/(e_H-1)Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}. This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.     

Graphs with maximum size and lower bounded girth

For integers n≥4 and ν≥n+1, let ex(ν;{C₃,…,C_n}) denote the maximum number of edges in a graph of order ν and girth at least n+1. The {C₃,…,C_n}-free graphs with order ν and size ex(ν;{C₃,…,C_n}) are called extremal graphs and denoted by EX(ν;{C₃,…,C_n}). We prove that given an integer k≥0, for each n≥2log₂(k+2) there exist extremal graphs with ν vertices, ν+k edges and minimum degree 1 or 2. Considering this idea we construct four infinite families of extremal graphs. We also see that minimal (r;g)-cages are the exclusive elements in EX(ν₀(r,g);{C₃,…,C_g−1}).     

Homogeneous Embeddings of Cycles in Graphs

If G and H are vertex-transitive graphs, then the framing number fr(G,H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates fr(C _m,C _n) for m<n. We show first that fr(C _m,C _n)≥n+2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that fr(C _m,C _n) is approximately the minimum of and n+n/m. Received: June 12, 1996 / Revised: June 2, 1997     

On linguistic dynamical systems,families of graphs of large girth,and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f _α : K ⁿ → K ⁿ depending on “time” α ∈ {K − 0} such that f _α ⁻¹ = f _−αM, the relation f _α1 (x) = f _α2 (x) for some x ∈ Kⁿ implies α₁ = α₂, and each map f _α has no invariant points. The neighborhood {f _α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kⁿ. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α₁, υ, α₂), s ≤ d, where α_i + α_i+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ _a = f _α1 × ⋯ × f _αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems L_n(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system L_n(q). If K = F_q, the graphs L(n, F_q) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems L_n(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.     

Adjacency matrices of polarity graphs and of other C 4-free graphs of large size

In this paper we give a method for obtaining the adjacency matrix of a simple polarity graph G _q from a projective plane PG(2, q), where q is a prime power. Denote by ex(n; C ₄) the maximum number of edges of a graph on n vertices and free of squares C ₄. We use the constructed graphs G _q to obtain lower bounds on the extremal function ex(n; C ₄), for some n < q ² + q + 1. In particular, we construct a C ₄-free graph on ${n=q^2 -\sqrt{q}}$ vertices and ${\frac{1}{2} q(q^2-1)-\frac{1}{2}\sqrt{q} (q-1) }$ edges, for a square prime power q.     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

On the total domination subdivision numbers in graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sd_{γ t} (G) ≤ 2γ _t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sd_{γ t} (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sd_{γ t} (G)=2γ _t (G)−1.     

Total Domination in Partitioned Graphs

We present results on total domination in a partitioned graph G = (V, E). Let γ _t (G) denote the total dominating number of G. For a partition , k ≥ 2, of V, let γ _t (G; V _i ) be the cardinality of a smallest subset of V such that every vertex of V _i has a neighbour in it and define the following

Edge Maximal Non-Bipartite Graphs Without Odd Cycles of Prescribed Lengths

 Let ?(n;3,5,…,2k+1) denote the class of non-bipartite graphs on n vertices having no odd cycle of length ≤2k+1. We prove that for every G∈?(n;3,5,…,2k+1) and characterize the extremal graphs. We also study the subclass ℋ(n;3,5,…,2k+1) consisting of the hamiltonian members of ?(n;3,5,…, 2k+1). For this subclass the above upper bound holds for odd n. For even n we establish the following sharp upper bound:

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999     

On graphs determining links with maximal number of components via medial construction

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K₁ by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

On the homogeneous algebraic graphs of large girth and their applications

Families of finite graphs of large girth were introduced in classical extremal graph theory. One important theoretical result here is the upper bound on the maximal size of the graph with girth ?2d established in Even Circuit Theorem by P. Erdös. We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth ?2d is established. Getting the lower bound, we use the family of bipartite graphs D(n,K) with n?2 over a field K, whose partition sets are two copies of the vector space Kⁿ. We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory. Finally, we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography.     

The smallest degree sum that yields potentially Kr,r-graphic sequences

We consider a variation of a classical Turán-type extremal problem as follows: Determine the smallest even integer σ(K_r,r,n) such that every n-term graphic sequence π = (d₁,d₂,...,d_n) with term sum σ(π) = d₁ + d₂ + ... + d_n ≥ σ(K_r,r,n) is potentially K_r,r-graphic, where K_r,r is an r × r complete bipartite graph, i.e. π has a realization G containing K_r,r as its subgraph. In this paper, the values σ(K_r,r,n) for even r and n ≥ 4r² - r - 6 and for odd r and n ≥ 4r² + 3r - 8 are determined.     

Poisson approximation for non-backtracking random walks

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [3] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1+o(1)))log n/log log n, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun’s sieve with some extensions of the ideas in [3]. Let N _t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N _t /n is typically 1/et! + o(1). Furthermore, if g = Ω(log log n), then N _t /n is typically 1+o(1)/et! niformly on all t ≤ (1 − o(1)) log n/log log n and 0 for all t ≥ (1 + o(1)) log n/log log n. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.     

Precise Asymptotics for Lévy Processes

Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX ²(1) < ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t ≥ 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables. This work is supported by the National Natural Science Foundation (Grant No. 10671188) and Special Foundation of USTC     

The Crossing Number of Cartesian Products of Complete Bipartite Graphs K 2, m with Paths P n

Most results on the crossing number of a graph focus on the special graphs, such as Cartesian products of small graphs with paths P_n, cycles C_n or stars S_n. In this paper, we extend the results to Cartesian products of complete bipartite graphs K_2,m with paths P_n for arbitrary m ≥ 2 and n ≥ 1. Supported by the NSFC (No. 10771062) and the program for New Century Excellent Talents in University.     

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 C_n 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 ≠ C_n} 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.