Polynomial time algorithms for recognizing and isomorphism testing of cyclic tournaments

In [3] the problem of finding an efficient criterion for isomorphism testing of cyclic graphs was posed. In the context of the theory of computational complexity the problem reduces to that of the existence of a polynomial-time algorithm for recognizing their isomorphism. The main result of the present paper is an algorithm for finding among all tournaments the cyclic ones. For cyclic tournaments generators of the automorphism group and the set of canonical labels are constructed. The running time of the algorithm is bounded by a polynomial function of the number of input tournament vertices. Thus an affirmative answer to the above problem is obtained.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Feedback Vertex Sets in Tournaments

We study combinatorial and algorithmic questions around minimal feedback vertex sets (FVS) in tournament graphs. On the combinatorial side, we derive upper and lower bounds on the maximum number of minimal FVSs in an n‐vertex tournament. We prove that every tournament on n vertices has at most 1.6740ⁿ minimal FVSs, and that there is an infinite family of tournaments, all having at least 1.5448ⁿ minimal FVSs. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal FVSs of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum‐sized FVS in a tournament.     

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.     

Finding the automorphism group of a circulant association scheme in polynomial time

A polynomial-time algorithm for finding the automorphism group of a circulant association scheme is constructed. The correctness of the algorithm is based on a new result generalizing the Burnside-Schur theorem (on permutation groups having a regular cyclic subgroup) in the class of automorphism groups of association schemes. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 251–267.     

Groupes de permutations primitifs d'ordre impair et denombrement de tournois sommet-primitifs

A tournament T (directed graph in which there is exactly one arc between any two vertices) is said to be point-primitive if its automorphism group A(T) acts primitively on the vertices of T. Automorphism groups of point-primitive tournaments are primitive permutation groups of odd order, which are known to be affine groups; and so point-primitive tournaments are of prime-power order. Some properties of primitive permutation groups of odd order are proved and a counting formula for point-primitive tournaments of order p^2k (p prime number, k integer ?0) is derived from them.     

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free.

In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.     

All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism

The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements. M. Giudici is supported by an Australian Postdoctoral Fellowship J. Xu was supported by an IPRS scholarship of Australia.     

Search algorithm for Ramsey graphs by union of group orbits

An algorithm for the construction of Ramsey graphs with a given automorphism group G is presented. To find a graph on n vertices with no clique of k vertices, K_k, and no independent set of l vertices, ¯K_l, k, l ≤ n, with an automorphism group G, a Boolean formula α based on the G-orbits of k-subsets and l-subsets of vertices is constructed from incidence matrices belonging to G. This Boolean formula is satisfiable if and only if the desired graph exists, and each satisfying assignment to α specifies a set of orbits of pairs of vertices whose union gives the edges of such a graph. Finding these assignments is basically equivalent to the conversion of α from CNF to DNF (conjunctive to disjunctive normal form). Though the latter problem is NP-hard, we present an “efficient” method to do the conversion for the formulas that appear in this particular problem. When G is taken to be the dihedral group D_n for n ≤ 101, this method matches all of the previously known cyclic Ramsey graphs, as reported by F. R. K. Chung and C. M. Grinstead [“A Survey of Bounds for Classical Ramsey Numbers,” Journal of Graph Theory, 7 (1983), 25–38], in dramatically smaller computer time when compared to the time required by an exhaustive search. Five new lower bounds for the classical Ramsey numbers are established: R(4, 7) ? 47, R(4, 8) ? 52, R(4, 9) ? 69, R(5,7) ? 76, and R(5, 8) ? 94. Also, some previously known cyclic graphs are shown to be unique up to isomorphism.     

Circulant tournaments of prime order are tight

We say that a tournament is tight if for every proper 3-coloring of its vertex set there is a directed cyclic triangle whose vertices have different colors. In this paper, we prove that all circulant tournaments with a prime number p≥3 of vertices are tight using results relating to the acyclic disconnection of a digraph and theorems of additive number theory.     

Constructing even radius tightly attached half-arc-transitive graphs of valency four

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41–76, 1998) classified odd radius tightly attached tetravalent half-arc-transitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K _4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.      

On graphs with regular groups

A new method for the construction of graphs with given regular group is developed and used to show that to every nonabelian group G of order 3^k, k ≥ 4, there exists a graph X whose automorphism group is isomorphic to G and regular as a permutation group on the set of vertices of X.     

Each 3-strong Tournament Contains 3 Vertices Whose Out-arcs Are Pancyclic

An arc in a tournament T with n ≥ 3 vertices is called pancyclic, if it is in a cycle of length k for all 3 ≤ k ≤ n. Yeo (Journal of Graph Theory, 50 (2005), 212–219) proved that every 3-strong tournament contains two distinct vertices whose all out-arcs are pancyclic, and conjectured that each 2-strong tournament contains 3 such vertices. In this paper, we confirm Yeo’s conjecture for 3-strong tournaments. The author is an associate member of “Graduiertenkolleg: Hierarchie und Symmetrie in mathematischen Modellen (DFG)” at RWTH Aachen University, Germany.     

Exotic n-universal graphs

An n-universal graph is a graph that contains as an induced subgraph a copy of every graph on n vertices. It is shown that for each positive integer n > 1 there exists an n-universal graph G on 4ⁿ - 1 vertices such that G is a (v, k, λ)-graph, and both G and its complement G¯ are 1-transitive in the sense of W. T. Tutte and are of diameter 2. The automorphism group of G is a transitive rank 3 permutation group, i.e., it acts transitively on (1) the vertices of G, (2) the ordered pairs uv of adjacent vertices of G, and (3) the ordered pairs xy of nonadjacent vertices of G.     

On Path Factors of (3, 4)-Biregular Bigraphs

A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.     

On a New High Dimensional Weisfeiler-Lehman Algorithm

We investigate the following problem: how different can a cellular algebra be from its Schurian closure, i.e., the centralizer algebra of its automorphism group? For this purpose we introduce the notion of a Schurian polynomial approximation scheme measuring this difference. Some natural examples of such schemes arise from high dimensional generalizations of the Weisfeiler-Lehman algorithm which constructs the cellular closure of a set of matrices. We prove that all of these schemes are dominated by a new Schurian polynomial approximation scheme defined by the m-closure operators. A sufficient condition for the m-closure of a cellular algebra to coincide with its Schurian closure is given.     

Extendability of Cyclic Orders

A cyclic order is a ternary relation that satisfies ternary transitivity and asymmetry conditions. Such a ternary relation is extendable if it is included in a complete cyclic order on the same ground set. Unlike the case of linear extensions of partial orders, a cyclic order need not be extendable. The extension problem for cyclic orders is to determine if a cyclic order is extendable. This problem is known to be NP-complete. We introduce a class of cyclic orders in which the extension problem can be solved in polynomial time. The class provides many explicit examples of nonextendable cyclic orders that were not previously known, including a nonextendable cyclic order on seven points. Let μ be the maximum cardinality of a ground set on which all cyclic orders are extendable. It has been shown that μ≤9. We prove that μ=6. This answers a question of Novák. In addition, we characterize the nonextendable cyclic orders on seven and eight points. Our results are intimately related to irreducible partially ordered set of order dimension three, and to fractional vertices of generalized transitive tournament polytopes. As by-products, we obtain a characterization of cyclically ordered sets of dimension two, and a new proof of a theorem of Dridi on small linear ordering polytopes. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Characterization of b‐Perfect Graphs

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     

The Linear Ordering Problem with cumulative costs

The optimization problem of finding a permutation of a given set of items that minimizes a certain cost function is naturally modeled by introducing a complete digraph G whose vertices correspond to the items to be sorted. Depending on the cost function to be used, different optimization problems can be defined on G. The most familiar one is the min-cost Hamiltonian path problem (or its closed-path version, the Traveling Salesman Problem), arising when the cost of a given permutation only depends on consecutive node pairs. A more complex situation arises when a given cost has to be paid whenever an item is ranked before another one in the final permutation. In this case, a feasible solution is associated with an acyclic tournament (the transitive closure of an Hamiltonian path), and the resulting problem is known as the Linear Ordering Problem (LOP).     

The number of vertices whose out-arcs are pancyclic in a 2-strong tournament

An arc going out from a vertex x in a digraph is called an out-arc of x. Yao et al. [Discrete Appl. Math. 99 (2000) 245-249] proved that every strong tournament contains a vertex x such that all out-arcs of x are pancyclic. Recently, Yeo [J. Graph Theory 50 (2005) 212-219] proved that each 3-strong tournament contains two such vertices. In this paper, we confirm that Yeo's result is also true for 2-strong tournaments. Our proof implies a polynomial algorithm to find two such vertices.