Strong Tournament Research Articles

Hamiltonian cycles avoiding prescribed arcs in semicomplete digraphs

Let f(k) (g(k), resp.) be the minimum positive integer such that every f(k)-strong semicomplete digraph (g(k)-strong tournament, resp.) contains a hamiltonian cycle avoiding any prescribed set of k arcs. Bang-Jensen et al. (2023) conjectured f(k)=k+2, which was inspired by the result g(k)=k+1 of Fraisse and Thomassen. Bang-Jensen and Jordán (2010) proved that every 3-strong semicomplete digraph contains a spanning 2-strong tournament. Guo (1997) proved that every (3k+1)-strong semicomplete digraph contains a spanning (k+1)-strong tournament. Combining with Fraisse and Thomassen’s result, Bang-Jensen, Havet and Yeo’s conjecture holds for k=1 and f(k)≤3k+1. In this paper, we improve the upper bound and prove f(k)≤2k+1 and the conjecture holds for k=2,3. In addition, we show that each (k+2)-strong semicomplete digraph has a hamiltonian cycle avoiding any k-arc set as long as each of its components has at most k−q2+2 vertices, where q is the number of components of the k-arc set.

Common Kings of a Chain of Cycles in a Strong Tournament

Common Kings of a Chain of Cycles in a Strong Tournament

Spanning acyclic subdigraphs and strong t-panconnectivity of tournaments

In 1980, C. Thomassen ( J. Combin. Theory Ser. B 28: 142-163) proved that if a 2-strong semicomplete digraph D contains three internally disjoint ( x , y ) -paths each of length at least 2, then D has a Hamiltonian ( x , y ) -path. It follows that every 4-strong tournament T is strongly Hamiltonian-connected. In this paper, we prove that if a semicomplete digraph D has a spanning acyclic subdigraph D ′ from x to y such that the length of the longest ( x , y ) -paths in D ′ is k , then D contains all ( x , y ) -paths of lengths at least k . By using this new sufficient condition to confirm the strong t -panconnectivity in D , we prove that if a 2-strong tournament D with n vertices contains m ≥ 3 internally disjoint ( x , y ) -paths each of length at least 2, then D has all ( x , y ) -paths of lengths at least n − 2 m + 3 . This implies that every ( m + 1 ) -strong tournament is strongly ( n − 2 m + 3 ) -panconnected.

The minimum number of spanning paths in a strong tournament

The minimum number of spanning paths in a strong tournament

Every (13k − 6)-strong tournament with minimum out-degree at least 28k − 13 is k-linked

A digraph D is k-linked if it satisfies that for every choice of disjoint sets {x1,…,xk} and {y1,…,yk} of vertices of D there are vertex disjoint paths P1,…,Pk such that Pi is an (xi,yi)-path. Confirming a conjecture by Kühn et al., Pokrovskiy proved in 2015 that every 452k-strong tournament is k-linked and asked for a better linear bound. Very recently Meng et al. proved that every (40k−31)-strong tournament is k-linked. In this note we use an important lemma from their paper to give a short proof that every (13k−6)-strong tournament of minimum out-degree at least 28k−13 is k-linked.

Eyes on the Prize: Do Industry Tournament Incentives Shape the Structure of Executive Compensation?

AbstractWe investigate whether external industry tournament incentives influence the design of executive-compensation contracts. Using staggered negative mobility shocks as exogenous disruptions to tournament incentives, we show that firms treated by these shocks act to restore their executives’ diminished implicit risk-taking incentives by increasing compensation vega. On average, post-shock compensation vegas increase by approximately 10%. These effects are considerably larger for treated executives with strong tournament incentives and high ex ante mobility. Mobility shocks have no impact on compensation delta or total pay. Our results shed light on how explicit risk-taking incentives are optimized with respect to executive career concerns.

Tournaments and Bipartite Tournaments without Vertex Disjoint Cycles of Different Lengths

M. A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math., 26 (2012), pp. 687--694] that a bipartite digraph of minimum out-degree at least 3 contains two vertex disjoint directed cycles of different lengths. In this paper, we disprove this conjecture. Further, we classify strong tournaments and strong bipartite tournaments of minimum out-degree 3 without two vertex disjoint directed cycles of different lengths.

Proximity and remoteness in directed and undirected graphs

Let D be a strongly connected digraph. The average distance σ̄(v) of a vertex v of D is the arithmetic mean of the distances from v to all other vertices of D. The remoteness ρ(D) and proximity π(D) of D are the maximum and the minimum of the average distances of the vertices of D, respectively. We obtain sharp upper and lower bounds on π(D) and ρ(D) as a function of the order n of D and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament T, we have π(T)=ρ(T) if and only if T is regular. Due to this result, one may conjecture that every strong digraph D with π(D)=ρ(D) is regular. We present an infinite family of non-regular strong digraphs D such that π(D)=ρ(D). We describe such a family for undirected graphs as well.

Spanning eulerian subdigraphs avoiding [formula omitted] prescribed arcs in tournaments

Fraise and Thomassen (1987) proved that every (k+1)-strong tournament has a hamiltonian cycle which avoids any prescribed set of k arcs. Bang-Jensen, Havet and Yeo showed in Bang-Jensen et al. (2019) that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian subdigraphs. They proved the existence of a smallest function f(k) with the property that every f(k)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs by showing that f(k)≤(k+1)2+44. They conjectured that every (k+1)-arc-strong semicomplete digraph has a spanning eulerian subdigraph which avoids any prescribed set of k arcs and verified this for k=2,3. In this paper we prove that f(k)≤⌈6k+15⌉. In particular, the conjecture holds for k≤4

Trail-connected tournaments

If for any two vertices v1 and v2 of digraph D, D admits a spanning (v1, v2)-dipath or a spanning (v2, v1)-dipath, then D is weakly Hamiltonian-connected; and if there are both a spanning (v1, v2)-dipath and a spanning (v2, v1)-dipath, then D is strongly Hamiltonian-connected. Thomassen in [J. of Combinatorial Theory, Series B, 28, (1980) 142–163] discovered a collection T0 of two digraphs and used it to characterize weakly Hamiltonian-connected tournaments, and proved that every 4-connected tournament is strongly Hamiltonian-connected. For any two vertices v1 and v2 of digraph D, D is weakly trail-connected if D admits a spanning (v1, v2)-ditrail or a spanning (v2, v1)-ditrail, and D is strongly trail-connected if there are both a spanning (v1, v2)-ditrail and a spanning (v2, v1)-ditrail. We have determined a family T of tournaments and prove the following. (i) A tournament D is weakly trail-connected if and only if D is strong. (ii) A strong tournament D is strongly trail-connected if and only if D is not a member in T. (iii) Every tournament with arc-strong connectivity at least 2 is strongly trail-connected.

Counting acyclic and strong digraphs by descents

A descent of a labeled digraph is a directed edge (s,t) with s>t. We count strong tournaments, strong digraphs, acyclic digraphs, and forests by descents and edges. To count strong tournaments we use Eulerian generating functions and to count strong and acyclic digraphs we use a new type of generating function that we call a graphic Eulerian generating function.

Eyes on the Prize: Do Industry Tournament Incentives Shape the Structure of Executive Compensation?

We investigate whether external industry tournament incentives influence the design of executive compensation contracts. Using staggered negative mobility shocks as exogenous disruptions to tournament incentives, we show that firms treated by these shocks act to restore their executives’ diminished implicit risk-taking incentives by increasing compensation vega. On average, post-shock compensation vegas increase by about 10%. These effects are considerably larger for treated executives with strong tournament incentives and high ex-ante mobility. Mobility shocks have no impact on compensation delta or total pay. Our results shed light on how explicit risk-taking incentives are optimized with respect to executive career concerns.

On the Strong n-partite Tournaments with Exactly Two Cycles of Length n − 1

Gutin and Rafiey (Australas J. Combin. 34 (2006), 17-21) provided an example of an n-partite tournament with exactly n − m + 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question. Let 3 ≤ m ≤ n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n − m + 1 cycles of length m for two values of m? In the same paper, they showed that this question has a negative answer for two values n − 1 and n. In this paper, we prove that a strong n-partite tournament with exactly two cycles of length n − 1 must contain some given multipartite tournament as subdigraph. As a corollary, we also show that the above question has a negative answer for two values n − 1 and any l with 3 ≤ l ≤ n and l ≠ n − 1.

Degree constrained 2-partitions of semicomplete digraphs

A 2-partition of a digraph D is a partition (V1,V2) of V(D) into two disjoint non-empty sets V1 and V2 such that V1∪V2=V(D). A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following:•For all integers k1,k2≥1 and k1+k2≥3 it is NP-complete to decide whether a given digraph D has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed choice of integers α,k1,k2≥1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition (V1,V2) such that D〈Vi〉 has out-degree at least ki for i=1,2.•For every fixed integer k≥1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition (V1,V2) such that D〈V1〉 has out-degree at least one and D〈V2〉 has in-degree at least k.•It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition (V1,V2) such that D〈Vi〉 is a strong tournament.

On the spanning connectivity of tournaments

Let D be a digraph. A k-container of D between u and v, C(u,v), is a set of k internally disjoint paths between u and v. A k-container C(u,v) of D is a strong (resp. weak) k∗-container (k≥2) if there is a set of k internally disjoint paths with the same direction (resp. with different directions allowed) between u and v and it contains all vertices of D. A digraph D is k∗-strongly (resp. k∗-weakly) connected if there exists a strong (resp. weak) k∗-container between any two distinct vertices for k≥2. Specially, we define D is 1∗-connected if D is weakly Hamiltonian connected (a 1∗-connected digraph is 1∗-strongly and also 1∗-weakly connected.) We define the strong (resp. weak) spanning connectivity of a digraph D, κs∗(D) (resp. κw∗(D) ), to be the largest integer k such that D is ω∗-strongly (resp. ω∗-weakly) connected for all 1≤ω≤k. In this paper, we show that for k≥0, a (2k+1)-strong tournament is (k+2)∗-weakly connected and that for k≥2, a 2k-strong tournament is k∗-strongly connected. Furthermore, we show that in a tournament with n vertices and irregularity i(T)≤k, if n≥6t+5k(t≥2), then κs∗(T)≥t and if n≥6t+5k−3(t≥2), then κw∗(T)≥t+1.

Strong rainbow connection in digraphs

An arc-coloured digraph is strongly rainbow connected if for every pair of vertices (u,v) there exists a shortest path from u to v all of whose arcs have different colours. The strong rainbow connection number of a digraph is the minimum number of colours needed to make the graph strongly rainbow connected.In this paper, we study the strong rainbow connection number of minimally strongly connected digraphs, non-Hamiltonian strong digraphs and strong tournaments.

Arc-disjoint hamiltonian paths in non-round decomposable local tournaments

Thomassen proved that a strong tournament T has a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices if and only if T is not an almost transitive tournament of odd order, where an almost transitive tournament is obtained from a transitive tournament with acyclic ordering u1,u2,…,un (i.e., ui→uj for all 1≤i<j≤n) by reversing the arc u1un. A digraph D is a local tournament if for every vertex x of D, both the out-neighbors and the in-neighbors of x induce tournaments. Bang-Jensen, Guo, Gutin and Volkmann split local tournaments into three subclasses: the round decomposable; the non-round decomposable which are not tournaments; the non-round decomposable which are tournaments. In 2015, we proved that every 2-strong round decomposable local tournament has a Hamiltonian path and a Hamiltonian cycle which are arc-disjoint if and only if it is not the second power of an even cycle. In this paper, we discuss the arc-disjoint Hamiltonian paths in non-round decomposable local tournaments, and prove that every 2-strong non-round decomposable local tournament contains a pair of arc-disjoint Hamiltonian paths with distinct initial vertices and distinct terminal vertices. This result combining with the one on round decomposable local tournaments extends the above-mentioned result of Thomassen to 2-strong local tournaments.

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with nge 2), there is a natural construction of a semigroup of transformations langle Drangle . For any edge (a, b) of D, let arightarrow b be the idempotent of rank n-1 mapping a to b and fixing all vertices other than a; then, define langle Drangle to be the semigroup generated by a rightarrow b for all (a,b) in E(D). For alpha in langle Drangle , let ell (D,alpha ) be the minimal length of a word in E(D) expressing alpha . It is well known that the semigroup mathrm {Sing}_n of all transformations of rank at most n-1 is generated by its idempotents of rank n-1. When D=K_n is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate ell (K_n,alpha ), for any alpha in langle K_nrangle = mathrm {Sing}_n; however, no analogous non-trivial results are known when D ne K_n. In this paper, we characterise all simple digraphs D such that either ell (D,alpha ) is equal to Howie–Iwahori’s formula for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {fix}(alpha ) for all alpha in langle Drangle , or ell (D,alpha ) = n - mathrm {rk}(alpha ) for all alpha in langle Drangle . We also obtain bounds for ell (D,alpha ) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank n-1 of mathrm {Sing}_n). We finish the paper with a list of conjectures and open problems.

On pancyclic arcs in hypertournaments

A k-hypertournament H on n vertices with 2≤k≤n is a pair H=(V,AH), where V is a set of n vertices and AH is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V, AH contains exactly one of the k!k-tuples whose entries belong to S. Obviously, a 2-hypertournament is a tournament.Moon (1994) proved that for every strong tournament, there is a Hamiltonian cycle which contains at least three pancyclic arcs. In this paper, we will show that for an arbitrary strong k-hypertournament H with n vertices, where 2≤k≤n−2, there is a Hamiltonian cycle C containing at least three pancyclic arcs, each of which belongs to an m-cycle Cm for each m∈{3,4,…,n} such that V(C3)⊂V(C4)⊂⋯⊂V(Cn).

Dicycle Cover of Hamiltonian Oriented Graphs

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.