Showing papers in "Journal of Graph Theory in 2012"

Vizing's conjecture: a survey and recent results

Abstract: Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012

On Meyniel's conjecture of the cop number

Abstract: Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper-bound O(n/ lnn) due to Chiniforooshan. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; contract grant numbers: DMS 0701111; DMS 1000475 (to L. L. and X. P.).)

Rainbow connection number and connected dominating sets

Abstract: The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from Schiermeyer (Combinatorial Algorithms, Springer, Berlin/Hiedelberg, 2009, pp. 432–437), improving the previously best known bound of 20n/δ (J Graph Theory 63 (2010), 185–191). This bound is tight up to additive factors by a construction mentioned in Caro et al. (Electr J Combin 15(R57) (2008), 1). As an intermediate step we obtain an upper bound of 3n/(δ + 1) − 2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree δ. This bound is tight up to an additive constant of 2. This result may be of independent interest. We also show that for every connected graph G with minimum degree at least 2, the rainbow connection number, rc(G), is upper bounded by Γc(G) + 2, where Γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G) + c, 1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, asteroidal triple-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G)⩽3·radius(G). In most of these cases, we also demonstrate the tightness of the bounds. © 2012 Wiley Periodicals, Inc.

Variations on Cops and Robbers

Abstract: We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≥1 edges at a time, establishing a general upper bound of , where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. We also show that in this case, the cop number of an n-vertex graph can be as large as n1 − 1/(R − 2) for finite R≥5, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)2/logn). Our approach is based on expansion. © 2011 Wiley Periodicals, Inc. J Graph Theory. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; Contract grant number: DMS-0753472 (to A. F.); Contract grant sponsor: USA-Israel BSF; Contract grant number: 2006322 (to M. K.); Contract grant sponsor: Israel Science Foundation; Contract grant number: 1063/08 (to M. K.); Contract grant sponsor: Pazy memorial award (to M. K.).)

Characterization of Edge-Transitive 4-Valent Bicirculants

Abstract: A bicirculant is a graph admitting an automorphism with exactly two vertex-orbits of equal size. All non-isomorphic 4-valent edge-transitive bicirculants are characterized in this article. As a corollary, a characterization of 4-valent arc-transitive dihedrants is obtained. © 2011 Wiley Periodicals, Inc. J Graph Theory. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Agencija za raziskovalno dejavnost Republike Slovenije; Contract grant number: P1-0285; Contract grant sponsor: Slovenian-Hungarian Intergovernmental Scientific and Technological Cooperation Project; Contract grant number: SI-2/2007.)

Antimagic labelling of vertex weighted graphs

Abstract: Suppose G is a graph, k is a non-negative integer. We say G is k-antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted-k-antimagic if for any vertex weight function w: V→ℕ, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well-known conjecture asserts that every connected graph G≠K2 is 0-antimagic. On the other hand, there are connected graphs G≠K2 which are not weighted-1-antimagic. It is unknown whether every connected graph G≠K2 is weighted-2-antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic. We also prove that every connected graph G≠K2 on n vertices is weighted- ⌊3n/2⌋-antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.

Long cycles in subgraphs of (pseudo)random directed graphs

Abstract: We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c − o(1))n. Moreover, there is a subgraph G′′of G with (1/2 + γ − o(1))|E| edges that does not contain a cycle of length at least cn. © 2011 Wiley Periodicals, Inc. J Graph Theory, © 2012 Wiley Periodicals, Inc. (Contract grant sponsors: ERC (to I. B.); USA-Israel BSF; Israel Science Foundation; Contract grant numbers: 2006322; 1063/08 (to M. K.); Contract grant sponsors: NSF grant DMS-1101185; NSF CAREER award DMS-0812005; USA-Israeli BSF.)

Packing of graphic n -tuples

Abstract: An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v1, …, vn} in which the degree of vi is the ith entry of π. Graphic n-tuples (d, …, d) and (d, …, d) pack if there are edge-disjoint n-vertex graphs G1 and G2 such that d(vi) = d and d(vi) = d for all i. We prove that graphic n-tuples π1 and π2 pack if , where Δand δdenote the largest and smallest entries in π1 + π2 (strict inequality when δ = 1); also, the bound is sharp. Kundu and Lovasz independently proved that a graphic n-tuple π is realized by a graph with a k-factor if the n-tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge-disjoint 1-factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k⩽3. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; Contract grant numbers: DMS-0914815 (to S. G. H.); DGE-0742434 (to M. S. J.); Contract grant sponsor: NSA; Contract grant number: H98230-10-1-0363 (to D. B. W.).)

Rank-width of random graphs

Abstract: Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour [J Combin Theory Ser B 96 (2006), 514–528]. We investigate the asymptotic behavior of rank-width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p∈(0, 1) is a constant, then rw(G(n, p)) = ⌈n/3⌉−O(1), (ii) if , then rw(G(n, p)) = ⌈1/3⌉−o(n), (iii) if p = c/n and c>1, then rw(G(n, p))⩾rn for some r = r(c), and (iv) if p⩽c/n and c81, then rw(G(n, p))⩽2. As a corollary, we deduce that the tree-width of G(n, p) is linear in n whenever p = c/n for each c>1, answering a question of Gao [2006]. © 2011 Wiley Periodicals, Inc. J Graph Theory, © 2012 Wiley Periodicals, Inc. (Contract grant sponsors: Samsung Scholarship (to C. L.); National Research Foundation of Korea (NRF); Contract grant number: 2011-0001185 (to J. L. and S. O.); Contract grant sponsor: TJ Park Junior Faculty Fellowship (to S. O.).)

Locally s-distance transitive graphs

Abstract: We give a unified approach to analysing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s � 2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph, or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups.

A Characterization of b-Perfect Graphs

Abstract: A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b-perfect if and only if it does not contain as an induced subgraph a member of a certain list of 22 graphs. This entails the existence of a polynomial-time recognition algorithm and of a polynomial-time algorithm for coloring exactly the vertices of every b-perfect graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:95–122, 2012 © 2012 Wiley Periodicals, Inc.

Extensions of Gallai–Ramsey results

Abstract: Consider the graph consisting of a triangle with a pendant edge. We describe the structure of rainbow -free edge colorings of a complete graph and provide some corresponding Gallai–Ramsey results. In particular, we extend a result of Gallai to find a partition of the vertices of a rainbow -free colored complete graph with a limited number of colors between the parts. We also extend some Gallai–Ramsey results of Chung and Graham, Faudree et al. and Gyarfas et al. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.

Regular Graphs, Eigenvalues and Regular Factors

Abstract: In this article, we obtain a sufficient condition for the existence of regular factors in a regular graph in terms of its third largest eigenvalue. We also determine all values of k such that every r-regular graph with the third largest eigenvalue at most has a k-factor. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: National Natural Science Foundation of China; Contract grant number: 11101329; Contract grant sponsor: Fundamental Research Funds for the Central Universities.)

Multiply balanced edge colorings of multigraphs

Abstract: In this article, a theorem is proved that generalizes several existing amalgamation results in various ways. The main aim is to disentangle a given edge-colored amalgamated graph so that the result is a graph in which the edges are shared out among the vertices in ways that are fair with respect to several notions of balance (such as between pairs of vertices, degrees of vertices in both the graph and in each color class, etc.). The connectivity of color classes is also addressed. Most results in the literature on amalgamations focus on the disentangling of amalgamated complete graphs and complete multipartite graphs. Many such results follow as immediate corollaries to the main result in this article, which addresses amalgamations of graphs in general, allowing for example the final graph to have multiple edges. A new corollary of the main theorem is the settling of the existence of Hamilton decompositions of the family of graphs K(a1, …, ap;λ1, λ2); such graphs arise naturally in statistical settings. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.

Walks and Paths in Trees

Abstract: Recently Csikvari [Combinatorica 30(2) 2010, 125–137] proved a conjecture of Nikiforov concerning the number of closed walks on trees. Our aim is to extend this theorem to all walks. In addition, we give a simpler proof of Csikvari's result and answer one of his questions in the negative. Finally we consider an analogous question for paths rather than walks. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; Contract grant numbers: CCR-0225610; DMS-0505550; W911NF-06-1-0076 (to B. B.).)

Every 4-Colorable Graph With Maximum Degree 4 Has an Equitable 4-Coloring

Abstract: Chen et al., conjectured that for r≥3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr, r (for odd r) and Kr + 1. If true, this would be a joint strengthening of the Hajnal–Szemeredi theorem and Brooks' theorem. Chen et al., proved that their conjecture holds for r = 3. In this article we study properties of the hypothetical minimum counter-examples to this conjecture and the structure of “optimal” colorings of such graphs. Using these properties and structure, we show that the Chen–Lih–Wu Conjecture holds for r≤4. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:31–48, 2012 © 2012 Wiley Periodicals, Inc.

Seymour's Second Neighborhood Conjecture for Tournaments Missing a Generalized Star

Abstract: Seymour's Second Neighborhood Conjecture asserts that every digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. We prove its weighted version for tournaments missing a generalized star. As a consequence the weighted version holds for tournaments missing a sun, star, or a complete graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:89–94, 2012 © 2012 Wiley Periodicals, Inc.

Acyclic edge coloring of 2-degenerate graphs

Abstract: An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contains seriesparallel graphs, outerplanar graphs, non - regular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a'(G)<=Delta + 2, where Delta = Delta(G) denotes the maximum degree of the graph. We prove the conjecture for 2-degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2-degenerate graph with maximum degree ?, then a'(G)<=Delta + 1. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 68:1-27, 2011

On the multi‐colored Ramsey numbers of cycles

Abstract: For a graph L and an integer k≥2, Rk(L) denotes the smallest integer N for which for any edge-coloring of the complete graph KN by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdos conjectured that, for an odd cycle Cn on n vertices, They proved the case when k = 2 and also provided an upper bound Rk(Cn)≤(k+ 2)!n. Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k≥4, When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that Rk(Cn)≥(k−1)n−2k+ 4. Here we prove that if n is even, then © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 169-175, 2012

Large cliques or stable sets in graphs with no four-edge path and no five-edge path in the complement

Abstract: Erdős and Hajnal [Discrete Math 25 (1989), 37–52] conjectured that, for any graph H, every graph on n vertices that does not have H as an induced subgraph contains a clique or a stable set of size nɛ(H) for some ɛ(H)>0. The Conjecture 1. known to be true for graphs H with |V(H)|≤4. One of the two remaining open cases on five vertices is the case where H is a four-edge path, the other case being a cycle of length five. In this article we prove that every graph on n vertices that does not contain a four-edge path or the complement of a five-edge path as an induced subgraph contains either a clique or a stable set of size at least n1/6. © 2011 Wiley Periodicals, Inc. J Graph Theory (This research was performed while the author was at Columbia University. © 2012 Wiley Periodicals, Inc.)

Kempe Equivalence of Edge-Colorings

Abstract: It is proved that all 4-edge-colorings of a (sub)cubic graph are Kempe equivalent. This resolves a conjecture of the second author. In fact, it is found that the maximum degree Δ = 3 is a threshold for Kempe equivalence of (Δ+1)-edge-colorings, as such an equivalence does not hold in general when Δ = 4. One extra color allows a similar result in this latter case; however, namely, when Δ≤4 it is shown that all (Δ+2)-edge-colorings are Kempe equivalent. © 2011 Wiley Periodicals, Inc. J Graph Theory (Bojan Mohar is on leave from IMFM 8 FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. © 2012 Wiley Periodicals, Inc.)

Connectivity keeping trees in k -connected graphs

Abstract: We show that one can choose the minimum degree of a k-connected graph G large enough (independent of the vertex number of G) such that G contains a copy T of a prescribed tree with the property that G − V(T) remains k-connected. This was conjectured in [W. Mader, J Graph Theory 65 (2010), 61–69]. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 324–329, 2012 © 2012 Wiley Periodicals, Inc.

The potential of greed for independence

Abstract: The well-known lower bound on the independence number of a graph due to Caro (Technical Report, Tel-Aviv University, 1979) and Wei (Technical Memorandum, TM 81 - 11217 - 9, Bell Laboratories, 1981) can be established as a performance guarantee of two natural and simple greedy algorithms or of a simple randomized algorithm. We study possible generalizations and improvements of these approaches using vertex weights and discuss conditions on so-called potential functions pG: V(G)→ℕ0 defined on the vertex set of a graph G for which suitably modified versions of the greedy algorithms applied to G yield independent sets I with *. We provide examples of such potentials, which lead to bounds improving the bound due to Caro and Wei. Furthermore, suitably adapting the randomized algorithm we give a short proof of Thiele's lower bound on the independence number of a hypergraph (T. Thiele, J Graph Theory 30 (1999), 213–221). © 2012 Wiley Periodicals, Inc.

Convex Excess in Partial Cubes

Abstract: The convex excess ce(G) of a graph G is introduced as where the summation goes over all convex cycles of G. It is proved that for a partial cube G with n vertices, m edges, and isometric dimension i(G), inequality 2n−m−i(G)−ce(G)≤2 holds. Moreover, the equality holds if and only if the so-called zone graphs of G are trees. This answers the question from Bre r et al. [Tiled partial cubes, J Graph Theory 40 (2002) 91–103] whether partial cubes admit this kind of inequalities. It is also shown that a suggested inequality from Bre r et al. [Tiled partial cubes, J Graph Theory 40 (2002) 91–103] does not hold. Copyright © 2011 John Wiley & Sons, Ltd. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Ministry of Science of Slovenia; Contract grant number: P1-0297 (to S. K.); Contract grant sponsor: NSA (S. S.).)

Optimal acyclic edge-coloring of cubic graphs

Abstract: An acyclic edge-coloring of a graph is a proper edge-coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge-coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K4 or K3,3; the acyclic chromatic index of K4 and K3,3 is 5. This result has previously been published by Fiamcik, but his published proof was erroneous. © 2012 Wiley Periodicals, Inc.

A Planar LAC

Abstract: The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [Math Slovaca 30 (1980), 405–417] stated the Linear Arboricity Conjecture (LAC) that the linear arboricity of any simple graph of maximum degree Δ is either ⌈Δ/2⌉ or ⌈(Δ + 1)/2⌉. In [J. L. Wu, J Graph Theory 31 (1999), 129–134; J. L. Wu and Y. W. Wu, J Graph Theory 58(3) (2008), 210–220], it was proven that LAC holds for all planar graphs. LAC implies that for Δ odd, la(G) = ⌈Δ/2⌉. We conjecture that for planar graphs, this equality is true also for any even Δ⩾6. In this article we show that it is true for any even Δ⩾10, leaving open only the cases Δ = 6, 8. We present also an O(n logn) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when Δ⩾9. © 2010 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: Bilateral Project; Contract grant number: BI-PL/08-09-008 (to M. C., Ł. K., and B. L.); Contract grant sponsor: Polish Ministry of Science and Higher Education; Contract grant number: N206 355636 (to M. C. and Ł. K.); Contract grant sponsor: National Natural Science Foundation of China; Contract grant numbers: 10871119; 10971121; 10901097; 10631070; 11001055 (to J.-F. Hou and J.-L. Wu); Contract grant sponsor: European Union, European Social Fund (to B. L.).)

Random Graphons and a Weak Positivstellensatz

Abstract: In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a “sum of squares” in the algebra of partially labeled graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.

A Sufficient Condition for Planar Graphs

Abstract: A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this article we prove that every planar graph without 4-cycles and without intersecting triangles is acyclically 5-choosable. This improves the result in [M. Chen and W. Wang, Discrete Math 308 (2008), 6216–6225], which says that every planar graph without 4-cycles and without two triangles at distance less than 3 is acyclically 5-choosable. © 2011 Wiley Periodicals, Inc. J Graph Theory (Contract grant sponsors: French Embassy in Beijing (to M. C.); The University of Bordeaux I (to M. C.); The LaBRI (to M. C.); CROUS (to M. C.). © 2012 Wiley Periodicals, Inc.)

The Cartesian Product of Hypergraphs

Abstract: We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and Žerovnik for the case of graphs and introduces the notion of diagonal-free grids as a replacement of the chord-free 4-cycles that play a crucial role in the case of graphs. This leads to a generalization of relation Δ on the arc set, whose convex hull is shown to coincide with the product relation of the prime factorization. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.

Distributing vertices on Hamiltonian cycles

Abstract: Let G be a graph of order n and 3≤t≤n/4 be an integer. Recently, Kaneko and Yoshimoto [J Combin Theory Ser B 81(1) (2001), 100–109] provided a sharp δ(G) condition such that for any set X of t vertices, G contains a hamiltonian cycle H so that the distance along H between any two vertices of X is at least n/2t. In this article, minimum degree and connectivity conditions are determined such that for any graph G of sufficiently large order n and for any set of t vertices X⊆V(G), there is a hamiltonian cycle H so that the distance along H between any two consecutive vertices of X is approximately n/t. Furthermore, the minimum degree threshold is determined for the existence of a hamiltonian cycle H such that the vertices of X appear in a prescribed order at approximately predetermined distances along H. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 28–45, 2012