Proof of a Conjecture of Henning and Yeo on Vertex-Disjoint Directed Cycles

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

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Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree

Discrete Mathematics ◽

10.1016/j.disc.2010.06.024 ◽

2010 ◽

Vol 310 (19) ◽

pp. 2567-2570 ◽

Cited By ~ 6

Author(s):

Nicolas Lichiardopol

Keyword(s):

Vertex Disjoint ◽

Directed Cycles

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Vertex-Disjoint Subtournaments of Prescribed Minimum Outdegree or Minimum Semidegree: Proof for Tournaments of a Conjecture of Stiebi

International Journal of Combinatorics ◽

10.1155/2012/273416 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Nicolas Lichiardopol

Keyword(s):

Maximum Order ◽

Strongly Connected ◽

Vertex Disjoint ◽

Directed Cycles

It was proved (Bessy et al., 2010) that for r≥1, a tournament with minimum semidegree at least 2r−1 contains at least r vertex-disjoint directed triangles. It was also proved (Lichiardopol, 2010) that for integers q≥3 and r≥1, every tournament with minimum semidegree at least (q−1)r−1 contains at least r vertex-disjoint directed cycles of length q. None information was given on these directed cycles. In this paper, we fill a little this gap. Namely, we prove that for d≥1 and r≥1, every tournament of minimum outdegree at least ((d2+3d+2)/2)r−(d2+d+2)/2 contains at least r vertex-disjoint strongly connected subtournaments of minimum outdegree d. Next, we prove for tournaments a conjecture of Stiebitz stating that for integers s≥1 and t≥1, there exists a least number f(s,t) such that every digraph with minimum outdegree at least f(s,t) can be vertex-partitioned into two sets inducing subdigraphs with minimum outdegree at least s and at least t, respectively. Similar results related to the semidegree will be given. All these results are consequences of two results concerning the maximum order of a tournament of minimum outdegree d (of minimum semidegree d) not containing proper subtournaments of minimum outdegree d (of minimum semidegree d).

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On the star forest polytope for trees and cycles

RAIRO - Operations Research ◽

10.1051/ro/2018076 ◽

2019 ◽

Vol 53 (5) ◽

pp. 1763-1773

Author(s):

Meziane Aider ◽

Lamia Aoudia ◽

Mourad Baïou ◽

A. Ridha Mahjoub ◽

Viet Hung Nguyen

Keyword(s):

Undirected Graph ◽

Dominating Set ◽

Discrete Math ◽

Complete Characterization ◽

Facial Structure ◽

Maximum Weight ◽

Single Node ◽

End Node ◽

Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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A fast $$(2 + \frac{2}{7})$$-approximation algorithm for capacitated cycle covering

Mathematical Programming ◽

10.1007/s10107-021-01678-3 ◽

2021 ◽

Author(s):

Vera Traub ◽

Thorben Tröbst

Keyword(s):

Vehicle Routing Problem ◽

Covering Problem ◽

Post Processing ◽

Routing Problem ◽

Cycle Cover ◽

Disjoint Cycles ◽

Processing Step ◽

Number Of Cycles ◽

Vertex Disjoint ◽

Capacitated Vehicle

AbstractWe consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.

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Cycle partitions of regular graphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000553 ◽

2020 ◽

pp. 1-24

Author(s):

Vytautas Gruslys ◽

Shoham Letzter

Keyword(s):

Regular Graph ◽

Bipartite Graphs ◽

Disjoint Paths ◽

Regular Graphs ◽

Simple Construction ◽

Vertex Set ◽

Almost All ◽

Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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On Directed Versions of the Hajnal–Szemerédi Theorem

Combinatorics Probability Computing ◽

10.1017/s0963548315000036 ◽

2015 ◽

Vol 24 (6) ◽

pp. 873-928 ◽

Cited By ~ 6

Author(s):

ANDREW TREGLOWN

Keyword(s):

Directed Graph ◽

Minimum Degree ◽

Directed Graphs ◽

Perfect Matchings ◽

Large Order ◽

Vertex Disjoint ◽

Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

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