Vertex pancyclic in‐tournaments

An in‐tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k‐pancyclicity of in‐tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in‐tournament is vertex k‐pancyclic for k ≤ 5 and k ≥ n − 3. In the latter case, we even show that the in‐tournaments in consideration are fully (n − 3)‐extendable which means that every vertex belongs to a cycle of length n − 3 and that the vertex set of every cycle of length at least n − 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in‐tournament of order n with minimum degree greater than is vertex k‐pancyclic for 5 < k < n − 3, and we present a family of examples showing that this bound would be best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 84–104, 2001     

Pancyclic arcs and connectivity in tournaments

A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a tournament T if it is contained in a cycle of length l, for every 3 ≤ l ≤ |T|. Let p(T) denote the number of pancyclic arcs in a tournament T. In 4 , Moon showed that for every non‐trivial strong tournament T, p(T) ≥ 3. Actually, he proved a somewhat stronger result: for any non‐trivial strong tournament h(T) ≥ 3 where h(T) is the maximum number of pancyclic arcs contained in the same hamiltonian cycle of T. Moreover, Moon characterized the tournaments with h(T) = 3. All these tournaments are not 2‐strong. In this paper, we investigate relationship between the functions p(T) and h(T) and the connectivity of the tournament T. Let p_k(n) := min {p(T), T k‐strong tournament of order n} and h_k(n) := min{h(T), T k‐strong tournament of order n}. We conjecture that (for k ≥ 2) there exists a constant α_k> 0 such that p_k(n) ≥ α_kn and h_k(n) ≥ 2k+1. In this paper, we establish the later conjecture when k = 2. We then characterized the tournaments with h(T) = 4 and those with p(T) = 4. We also prove that for k ≥ 2, p_k(n) ≥ 2k+3. At last, we characterize the tournaments having exactly five pancyclic arcs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 87–110, 2004     

Vertex‐pancyclicity of hypertournaments

A hypertournament or a k‐tournament, on n vertices, 2≤k≤n, is a pair T=(V, E), where the vertex set V is a set of size n and the edge set E is the collection of all possible subsets of size k of V, called the edges, each taken in one of its k! possible permutations. A k‐tournament is pancyclic if there exists (directed) cycles of all possible lengths; it is vertex‐pancyclic if moreover the cycles can be found through any vertex. A k‐tournament is strong if there is a path from u to v for each pair of distinct vertices u and v. A question posed by Gutin and Yeo about the characterization of pancyclic and vertex‐pancyclic hypertournaments is examined in this article. We extend Moon's Theorem for tournaments to hypertournaments. We prove that if k≥8 and n≥k + 3, then a k‐tournament on n vertices is vertex‐pancyclic if and only if it is strong. Similar results hold for other values of k. We also show that when n≥7, k≥4, and n≥k + 2, a strong k‐tournament on n vertices is pancyclic if and only if it is strong. The bound n≥k+ 2 is tight. We also find bounds for the generalized problem when we extend vertex‐pancyclicity to require d edge‐disjoint cycles of each possible length and extend strong connectivity to require d edge‐disjoint paths between each pair of vertices. Our results include and extend those of Petrovic and Thomassen. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 338–348, 2010     

The structure of 4‐strong tournaments containing exactly three out‐arc pancyclic vertices

Yao et al. (Discrete Appl Math 99 (2000), 245–249) proved that every strong tournament contains a vertex u such that every out‐arc of u is pancyclic and conjectured that every k‐strong tournament contains k such vertices. At present, it is known that this conjecture is true for k = 1, 2, 3 and not true for k?4. In this article, we obtain a sufficient and necessary condition for a 4‐strong tournament to contain exactly three out‐arc pancyclic vertices, which shows that a 4‐strong tournament contains at least four out‐arc pancyclic vertices except for a given class of tournaments. Furthermore, our proof yields a polynomial algorithm to decide if a 4‐strong tournament has exactly three out‐arc pancyclic vertices.     

Tournois sans intervalle acyclique

An interval X of a tournament T is a vertex subset of T such that any vertex not in X either dominates or is dominated by all of the vertices in X. We caracterize the tournaments such that the only non empty acyclic intervals are the singletons and which are critical for that property, that is whenever a vertex is removed at least one acyclic interval with more than 2 vertices is created. These tournaments are exactly those which are the composition of any tournament with circulant tournaments. That work on acyclic intervals was motivated by the study of tournaments for which no median order forced itself naturally. To cite this article: J.-F. Culus, B. Jouve, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Orthogonal factorizations of graphs

Let G be a graph with vertex set V(G) and edge set E(G). Let k₁, k₂,…,k_m be positive integers. It is proved in this study that every [0,k₁+…+k_m?m+1]‐graph G has a [0, k_i]₁^m‐factorization orthogonal to any given subgraph H with m edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 267–276, 2002     

Finding Scores in Tournaments

A tournamentT_nis an orientation of the complete graph onnvertices. We continue the algorithmic study initiated by10of recognizing various directed trees in tournaments. Hell and Rosenfeld studied the complexity of finding various oriented paths in tournaments by probing edge directions. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree) in the same model. We show that the complexity of finding a vertex of outdegreek( ≤ (n − 1)/2) inT_nis Θ(nk). This bound is in sharp contrast to the Θ(n) bound for selection in the case of transitive tournaments. We also establish tight bounds for finding vertices of prescribed degree from the adjacency matrix of general directed/undirected graphs. These bounds generalize the classical bound of11for finding a sink (a vertex of outdegree 0 and indegreen − 1) in a directed graph.     

An application of the combinatorial Nullstellensatz to a graph labelling problem

An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,…,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K₂, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n=p^k vertices, where p is an odd prime and k is a positive integer that admits a C_p‐factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50 (2005), 263–272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1–2) (1999), 7–29]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 70–82, 2010.     

Cycle factors in strongly connected local tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called a local tournament if the set of in-neighbors as well as the set of out-neighbors of every vertex induces a tournament. A digraph is 2-connected if the removal of an arbitrary vertex results in a strongly connected digraph.In 2004 and 2005, Li and Shu investigated the structure of strongly connected, but not 2-connected tournaments. Using their structural results they were able to give sufficient conditions for a strongly connected tournament T to have complementary cycles or a k-cycle factor, i.e. a set of k vertex disjoint cycles that span the vertex set of T.Inspired by the articles of Li and Shu we develop in this paper the structure necessary for a strongly connected local tournament to be not cycle complementary. Using this structure, we are able to generalize and transfer various results of Li and Shu to the class of local tournaments.     

Edge‐disjoint Hamiltonian cycles in hypertournaments

We introduce a method for reducing k‐tournament problems, for k ≥ 3, to ordinary tournaments, that is, 2‐tournaments. It is applied to show that a k‐tournament on n ≥ k + 1 + 24d vertices (when k ≥ 4) or on n ≥ 30d + 2 vertices (when k = 3) has d edge‐disjoint Hamiltonian cycles if and only if it is d‐edge‐connected. Ironically, this is proved by ordinary tournament arguments although it only holds for k ≥ 3. We also characterizatize the pancyclic k‐tournaments, a problem posed by Gutin and Yeo.(Our characterization is slightly incomplete in that we prove it only for n large compared to k.). © 2005 Wiley Periodicals, Inc. J Graph Theory     

Ordered interval routing schemes

An Interval Routing Scheme (IRS) represents the routing tables in a network in a space-efficient way by labeling each vertex with an unique integer address, and the outgoing edges at each vertex with disjoint subintervals of these addresses. An IRS that has at most k intervals per edge label is called a k-IRS. In this paper, we propose a new type of interval routing scheme, called an Ordered Interval Routing Scheme (OIRS), that uses an ordering of the outgoing edges at each vertex and allows non-disjoint intervals in the labels of those edges. We show for a number of graph classes that using an OIRS instead of an IRS reduces the size of the routing tables in the case of optimal routing, i.e., routing along shortest paths. We show that optimal routing in any k-tree is possible using an OIRS with at most 2^k−1 intervals per edge label, although the best known result for an IRS is 2^k+1 intervals per edge label. Any torus has an optimal 1-OIRS, although it may not have an optimal 1-IRS. We present similar results for the Petersen graph, k-garland graphs and a few other graphs.     

Star-factors of tournaments

Let S_m denote the m-vertex simple digraph formed by m − 1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n/m disjoint copies of S_m. We prove that m lg m − m lg lg m ≤ f(m) ≤ 4m² − 6m for sufficiently large m. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 141–145, 1998     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.     

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K ₂, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each k 3 2, q 3 \binomk+12{k\geq 2,\,q\geq\binom{k+1}{2}} with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.     

Anti‐magic graphs via the Combinatorial NullStellenSatz

An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,…,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it has an antimagic labeling. In [ 10 ], Ringel conjectured that every simple connected graph, other than K₂, is antimagic. We prove several special cases and variants of this conjecture. Our main tool is the Combinatorial NullStellenSatz (cf. [ 1 ]). © 2005 Wiley Periodicals, Inc. J Graph Theory     

Edge intersection graphs of systems of paths on a grid with a bounded number of bends

We answer some of the questions raised by Golumbic, Lipshteyn and Stern and obtain some other results about edge intersection graphs of paths on a grid (EPG graphs). We show that for any d≥4, in order to represent every n vertex graph with maximum degree d as an edge intersection graph of n paths on a grid, a grid of area Θ(n²) is needed. We also show several results related to the classes B_k-EPG, where B_k-EPG denotes the class of graphs that have an EPG representation such that each path has at most k bends. In particular, we prove: For a fixed k and a sufficiently large n, the complete bipartite graph K_m,n does not belong to B_2m−3-EPG (it is known that this graph belongs to B_2m−2-EPG); for any odd integer k we have B_k-EPG B_k+1-EPG; there is no number k such that all graphs belong to B_k-EPG; only 2^O(knlog(kn)) out of all the labeled graphs with n vertices are in B_k-EPG.     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

Structure theorem for tournaments omitting N5

A finite tournament T is tight if the class of finite tournaments omitting T is well‐quasi‐ordered. We show here that a certain tournament N₅ on five vertices is tight. This is one of the main steps in an exact classification of the tight tournaments, as explained in [10]; the third and final step is carried out in [11]. The proof involves an encoding of the indecomposable tournaments omitting N₅ by a finite alphabet, followed by an application of Kruskal's Tree Theorem. This problem arises in model theory and in computational complexity in a more general form, which remains open: the problem is to give an effective criterion for a finite set {T₁,…,T_k} of finite tournaments to be tight in the sense that the class of all finite tournaments omitting each of T₁,…,T_k is well‐quasi‐ordered. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 165–192, 2003     

A note on bipartite subgraphs of triangle-free graphs

Let G be a triangle-free graph on n points with m edges and vertex degrees d₁, d₂,…, d_n. Let k be the maximum number of edges in a bipartite subgraph of G. In this note we show that k ? m/2 + Σ √d_i. It follows as a corollary that k ? m/2 + cm^3/4.     

A phase transition phenomenon in a random directed acyclic graph

Let a random directed acyclic graph be defined as being obtained from the random graph G_{n, p} by orienting the edges according to the ordering of vertices. Let γ_n* be the size of the largest (reflexive, transitive) closure of a vertex. For p=c(log n)/n, we prove that, with high probability, γ_n* is asymptotic to n^c log n, 2n(log log n)/log n, and n(1−1/c) depending on whether c<1, c=1, or c>1. We also determine the limiting distribution of the first vertex closure in all three ranges of c. As an application, we show that the expected number of comparable pairs is asymptotic to n^1+c/c log n, ½(n(log log n)/log n)², and ½(n(1−1/c))², respectively. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 164–184, 2001