On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

Highly connected coloured subgraphs via the regularity lemma

For integers , n≥k and r≥s, let m(n,r,s,k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph K_n on n vertices. Bollobás asked for the determination of m(n,r,s,k).Here, bounds are obtained in the cases s=1,2 and k=o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours.We shall also study a similar question for bipartite graphs.     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

On the rank spread of graphs

For a simple graph G?=?(𝒱, ?) with vertex-set 𝒱?=?{1,?…?,?n}, let 𝒮(G) be the set of all real symmetric n-by-n matrices whose graph is G. We present terminology linking established as well as new results related to the minimum rank problem, with spectral properties in graph theory. The minimum rank mr(G) of G is the smallest possible rank over all matrices in 𝒮(G). The rank spread r _v (G) of G at a vertex v, defined as mr(G)???mr(G???v), can take values ??∈?{0,?1,?2}. In general, distinct vertices in a graph may assume any of the three values. For ??=?0 or 1, there exist graphs with uniform r _v (G) (equal to the same integer at each vertex v). We show that only for ??=?0, will a single matrix A in 𝒮(G) determine when a graph has uniform rank spread. Moreover, a graph G, with vertices of rank spread zero or one only, is a λ-core graph for a λ-optimal matrix A in 𝒮(G). We also develop sufficient conditions for a vertex of rank spread zero or two and a necessary condition for a vertex of rank spread two.     

Labeling the -path with a condition at distance two

For integer r≥2, the infinite r-path P_∞(r) is the graph on vertices …v₋₃,v₋₂,v₋₁,v₀,v₁,v₂,v₃… such that v_s is adjacent to v_t if and only if |s−t|≤r−1. The r-path on n vertices is the subgraph of P_∞(r) induced by vertices v₀,v₁,v₂,…,v_n−1. For non-negative reals x₁ and x₂, a λ_x1,x2-labeling of a simple graph G is an assignment of non-negative reals to the vertices of G such that adjacent vertices receive reals that differ by at least x₁, vertices at distance two receive reals that differ by at least x₂, and the absolute difference between the largest and smallest assigned reals is minimized. With λ_x1,x2(G) denoting that minimum difference, we derive λ_x1,x2(P_n(r)) for r≥3, 1≤n≤∞, and . For , we obtain upper bounds on λ_x1,x2(P_∞(r)) and use them to give λ_x1,x2(P_∞(r)) for r≥5 and . We also determine λ_x1,x2(P_∞(3)) and λ_x1,x2(P_∞(4)) for all .     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

On the number of edge disjoint cliques in graphs of given size

In this paper, we prove that any graph ofn vertices andt _r–1(n)+m edges, wheret _r–1(n) is the Turán number, contains (1–o(1)m edge disjointK _r'sifm=o(n ²). Furthermore, we determine the maximumm such that every graph ofn vertices andt _r–1(n)+m edges containsm edge disjointK _r's ifn is sufficiently large.Research partially supported by Hungarian National Foundation for Scientific Research Grant no. 1812.     

A Metric of Constant Curvature on Polycycles

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions:(1) each internal face is an r-gon, where r ≥ 3;(2) the degree of each internal vertex is q, where q ≥ 3, and the degree of each boundary vertex is at most q and at least 2.Then it also possesses the following third property:(3) the vertices, the edges, and the internal faces form a cell complex.Simple examples show that conditions (1) and (2) are independent even provided condition (3) is satisfied. These are the defining conditions for an (r, q)-polycycle.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 223–233.Original Russian Text Copyright © 2005 by M. Deza, M. I. Shtogrin.     

On extremal problems of graphs and generalized graphs

Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n ₀ everyr-graph ofn vertices andn ^{r−ε(l, r)} r-tuples containsr. l verticesx ^(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples occur in ther-graph.     

Minimum degree of minimal defect -extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. If for any edge e in a defect n-extendable graph G, G−e is not defect n-extendable, then G is minimal defect n-extendable. The minimum degree and the connectivity of a graph G are denoted by δ(G) and κ(G) respectively. In this paper, we study the minimum degree of minimal defect n-extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has δ(G)=1. Consider a minimal defect n-extendable bipartite graph G with n≥2, we show that if κ(G)=1, then δ(G)≤n+1 and if κ(G)≥2, then 2≤δ(G)=κ(G)≤n+1. In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds.     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

On Score Sequences ofk-Hypertournaments

Given two nonnegative integers n and k withn ≥ k > 1, a k -hypertournament on n vertices is a pair (V, A), where V is a set of vertices with | V | = n and A is a set of k -tuples of vertices, called arcs, such that for any k -subset S ofV , A contains exactly one of the k!k -tuples whose entries belong to S. We show that a nondecreasing sequence (r₁, r₂, , r_n) of nonnegative integers is a losing score sequence of a k -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. We also show that a nondecreasing sequence (s₁,s₂ , , s_n) of nonnegative integers is a score sequence of somek -hypertournament if and only if for each j(1 ≤ j ≤ n),with equality holding whenj = n. Furthermore, we obtain a necessary and sufficient condition for a score sequence of a strong k -hypertournament. The above results generalize the corresponding theorems on tournaments.     

Conditions for -graphic sequences to be potentially -graphic

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially -graphic if it has a realization containing as a subgraph. In this paper, some conditions for r-graphic sequences to be potentially -graphic are given. These are generalizations from 1-graphs to r-graphs of four theorems due to Rao [A.R. Rao, The clique number of a graph with given degree sequence, in: A.R. Rao (Ed.), Proc. Symposium on Graph Theory, in: I.S.I. Lecture Notes Series, vol. 4, MacMillan and Co. India Ltd., (1979), 251–267; A.R. Rao, An Erdös-Gallai type result on the clique number of a realization of a degree sequence (unpublished)] and Kézdy and Lehel [A.E. Kézdy, J. Lehel, Degree sequences of graphs with prescribed clique size, in: Y. Alavi et al., (Eds.), in: Combinatorics, Graph Theory, and Algorithms, vol. 2, New Issues Press, Kalamazoo Michigan, 1999, 535–544].     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

On the Ramsey number of trees versus graphs with large clique number

Chvátal established that r(T_m, K_n) = (m – 1)(n – 1) + 1, where T_m is an arbitrary tree of order m and K_n is the complete graph of order n. This result was extended by Chartrand, Gould, and Polimeni who showed K_n could be replaced by a graph with clique number n and order n + 1 provided n ≧ 3 and m ≧ 3. We further extend these results to show that K_n can be replaced by any graph on n + 2 vertices with clique number n, provided n ≧ 5 and m ≧ 4. We then show that further extensions, in particular to graphs on n + 3 vertices with clique number n are impossible. We also investigate the Ramsey number of trees versus complete graphs minus sets of independent edges. We show that r(T_m, K_n –tK₂) = (m – 1)(n – t – 1) + 1 for m ≧ 3, n ≧ 6, where T_m is any tree of order m except the star, and for each t, O ≦ t ≦ [(n – 2)/2].     

Prophet inequalities for cost of observation stopping problems

Comparisons are made between the expected gain of a prophet (an observer with complete foresight) and the maximal expected gain of a gambler (using only non-anticipating stopping times) observing a sequence of independent, uniformly bounded random variables where a non-negative fixed cost is charged for each observation. Sharp universal bounds are obtained under various restrictions on the cost and the length of the sequence. For example, it is shown for X₁, X₂, … independent, [0, 1]-valued random variables that for all c ≥ 0 and all n ≥ 1 that E(max_{1 ≤ j ≤ n}(X_j − jc)) − sup_{t Tn} E(X_t − tc) ≤ 1/e, where T_n is the collection of all stopping times t which are less than or equal to n almost surely.     

The Ramsey numbers for cycles versus wheels of odd order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. It is conjectured by Surahmat, E.T. Baskoro and I. Tomescu that R(C_n,W_m)=2n−1 for even m≥4, n≥m and (n,m)≠(4,4). In this paper, we confirm the conjecture for n≥3m/2+1.     

Computing Vertex Connectivity: New Bounds from Old Techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ³ + n, κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ₁nmlog(n²/m)) where κ₁ ≤ m/n is the unweighted vertex connectivity or in expected time O(nmlog(n²/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow-push algorithm of Hao and Orlin (1994, J. Algorithms17, 424–446) that computes edge connectivity.     

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).     

The Ramsey numbers for cycles versus wheels of even order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. Surahmat, Baskoro and Tomescu conjectured that R(C_n,W_m)=3n−2 for m odd, n≥m≥3 and (n,m)≠(3,3). In this paper, we confirm the conjecture for n≥20.