scholarly journals Jones' Conjecture in Subcubic Graphs

10.37236/9192 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Marthe Bonamy ◽  
François Dross ◽  
Tomáš Masařík ◽  
Andrea Munaro ◽  
Wojciech Nadara ◽  
...  
Keyword(s):  
Planar Graph ◽  
Disjoint Cycles ◽  
Subcubic Graphs ◽  
Vertex Disjoint

We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to obtain a forest.

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

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.

Vertex disjoint cycles of different lengths in d-arc-dominated digraphs

2014 ◽  
Vol 42 (5) ◽  
pp. 351-354 ◽  
Author(s):  
Ngo Dac Tan
Keyword(s):  
Disjoint Cycles ◽  
Vertex Disjoint

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

Two vertex-disjoint cycles in a graph

1995 ◽  
Vol 11 (4) ◽  
pp. 389-396 ◽  
Author(s):  
Hong Wang
Keyword(s):  
Disjoint Cycles ◽  
Vertex Disjoint

scholarly journals Independence number and vertex-disjoint cycles

2007 ◽  
Vol 307 (11-12) ◽  
pp. 1493-1498 ◽  
Author(s):  
Yoshimi Egawa ◽  
Hikoe Enomoto ◽  
Stanislav Jendrol ◽  
Katsuhiro Ota ◽  
Ingo Schiermeyer
Keyword(s):  
Independence Number ◽  
Disjoint Cycles ◽  
Vertex Disjoint

Vertex-disjoint cycles containing prescribed vertices

2003 ◽  
Vol 42 (4) ◽  
pp. 276-296 ◽  
Author(s):  
Yoshiyasu Ishigami ◽  
Tao Jiang
Keyword(s):  
Disjoint Cycles ◽  
Vertex Disjoint

Cycle Transversals in Tournaments with Few Vertex Disjoint Cycles

2014 ◽  
Vol 79 (4) ◽  
pp. 249-266
Author(s):  
Jørgen Bang-Jensen ◽  
Stéphane Bessy
Keyword(s):  
Disjoint Cycles ◽  
Vertex Disjoint

scholarly journals Large independent sets in triangle-free cubic graphs: beyond planarity

2020 ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Gwenaël Joret ◽  
Jan Goedgebeur
Keyword(s):  
Planar Graph ◽  
Present Article ◽  
Additional Assumption ◽  
Independence Number ◽  
Independent Set ◽  
Cubic Graphs ◽  
Subcubic Graph ◽  
Color Class ◽  
Subcubic Graphs ◽  
Colorable Graph

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Charles Delorme ◽  
Guillermo Pineda-Villavicencio
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Minimum Degree ◽  
Integer Coefficients ◽  
Disjoint Cycles ◽  
The Matrix ◽  
Odd Degree ◽  
Unexplored Area ◽  
Trivial Graph ◽  
Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

scholarly journals Disjoint Cycles of Different Lengths in Graphs and Digraphs

10.37236/6921 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Julien Bensmail ◽  
Ararat Harutyunyan ◽  
Ngoc Khang Le ◽  
Binlong Li ◽  
Nicolas Lichiardopol
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Probabilistic Method ◽  
Undirected Graphs ◽  
Disjoint Cycles ◽  
Vertex Disjoint ◽  
Degree Bound ◽  
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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