644 research outputs found
Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments
A long-standing conjecture of Kelly states that every regular tournament on n
vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove
this conjecture for large n. In fact, we prove a far more general result, based
on our recent concept of robust expansion and a new method for decomposing
graphs. We show that every sufficiently large regular digraph G on n vertices
whose degree is linear in n and which is a robust outexpander has a
decomposition into edge-disjoint Hamilton cycles. This enables us to obtain
numerous further results, e.g. as a special case we confirm a conjecture of
Erdos on packing Hamilton cycles in random tournaments. As corollaries to the
main result, we also obtain several results on packing Hamilton cycles in
undirected graphs, giving e.g. the best known result on a conjecture of
Nash-Williams. We also apply our result to solve a problem on the domination
ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by
Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust
decomposition lemma' for application in subsequent paper
A Characterization of Connected (1,2)-Domination Graphs of Tournaments
Recently. Hedetniemi et aI. introduced (1,2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1,2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1,2)-dominating graph of D, dom1,2 (D), is defined to be the graph G = (V, E ) , where V (G) = V (D), and xy is an edge of G whenever x and y form a (1,2)-dominating pair in D. In this paper, we characterize all connected graphs that can be (I, 2)-dominating graphs of tournaments
The (1,2)-Step Competition Graph of a Tournament
The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the m-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)-step competition graph of a digraph D, denoted C1,2(D), as the graph on V(D) where {x,y}∈E(C1,2(D)) if and only if there exists a vertex z≠x,y, such that either dD−y(x,z)=1 and dD−x(y,z)≤2 or dD−x(y,z)=1 and dD−y(x,z)≤2. In this paper, we characterize the (1,2)-step competition graphs of tournaments and extend our results to the (i,k)-step competition graph of a tournament
Digraphs with Isomorphic Underlying and Domination Graphs: Pairs of Paths
A domination graph of a digraph D, dom (D), is created using thc vertex set of D and edge uv ϵ E (dom (D)) whenever (u, z) ϵ A (D) or (v, z) ϵ A (D) for any other vertex z ϵ A (D). Here, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. Specifically, digraphs are completely characterized where UGc (D) is the union of two disjoint paths
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
A digraph such that every proper induced subdigraph has a kernel is said to
be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI
for short) resp.) if the digraph has a kernel (does not have a kernel resp.).
The unique CKI-tournament is and the unique
KP-tournaments are the transitive tournaments, however bipartite tournaments
are KP. In this paper we characterize the CKI- and KP-digraphs for the
following families of digraphs: locally in-/out-semicomplete, asymmetric
arc-locally in-/out-semicomplete, asymmetric -quasi-transitive and
asymmetric -anti-quasi-transitive -free and we state that the problem
of determining whether a digraph of one of these families is CKI is polynomial,
giving a solution to a problem closely related to the following conjecture
posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for
locally in-semicomplete digraphs.Comment: 13 pages and 5 figure
Modularity and Optimality in Social Choice
Marengo and the second author have developed in the last years a geometric
model of social choice when this takes place among bundles of interdependent
elements, showing that by bundling and unbundling the same set of constituent
elements an authority has the power of determining the social outcome. In this
paper we will tie the model above to tournament theory, solving some of the
mathematical problems arising in their work and opening new questions which are
interesting not only from a mathematical and a social choice point of view, but
also from an economic and a genetic one. In particular, we will introduce the
notion of u-local optima and we will study it from both a theoretical and a
numerical/probabilistic point of view; we will also describe an algorithm that
computes the universal basin of attraction of a social outcome in O(M^3 logM)
time (where M is the number of social outcomes).Comment: 42 pages, 4 figures, 8 tables, 1 algorithm
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