Filters








2,298 Hits in 3.9 sec

Partitioning vertices of 1-tough graphs into paths

Cristina Bazgan, Amel Harkat-Benhamdine, Hao Li, Mariusz Woźniak
2001 Theoretical Computer Science  
In this paper we prove that every 1-tough graph has a partition of its vertices into paths of length at least two.  ...  Proof of Theorem 3 Suppose that G is a 1-tough graph with at least three vertices which does not have a partition of its vertices into a long path system.  ...  If G is a 1-tough graph with at least three vertices; then G has a partition of its vertices into a long path system. We will give a complete proof of this theorem in Section 3.  ... 
doi:10.1016/s0304-3975(00)00247-4 fatcat:dtvrxaoltjet7pj3oictw6kdv4

Hamiltonian powers in threshold and arborescent comparability graphs

Sam Donnelly, Garth Isaak
1999 Discrete Mathematics  
We examine powers of Hamiltonian paths and cycles as well as Hamiltonian (power) completion problems in several highly structured graph classes.  ...  For threshold graphs we give efficient algorithms as well as sufficient and minimax toughness like conditions.  ...  Acknowledgements Thanks to a referee for a very careful reading of the manuscript and useful comments.  ... 
doi:10.1016/s0012-365x(98)00346-x fatcat:llno563brrehdfdyt5s2jk6kei

Toughness, hamiltonicity and split graphs

Dieter Kratsch, Jenő Lehel, Haiko Müller
1996 Discrete Mathematics  
Related to Chvfital's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonieity on special classes of graphs.  ...  We show that every 3-tough split graph is hamiltonian and that there is a sequence of nonhamiltonian split graphs with toughness converging to 3.  ...  Let ff be the set of all partitions of W UB into B-paths and edges.  ... 
doi:10.1016/0012-365x(95)00190-8 fatcat:3vqxxzp2sfc5tlxhqv6xefmt3m

A Boundary Class for the k-Path Partition Problem

Nicholas Korpelainen
2018 Electronic Notes in Discrete Mathematics  
We establish the first known boundary class for the k-path partition problem and deduce that for a graph class defined by finitely many minimal forbidden induced subgraphs, the k-path partition problem  ...  remains NP-hard unless one of the forbidden induced subgraphs is a subcubic tree (a tree of maximum degree at most 3) with at most one vertex of degree 3.  ...  It remains to find a P k -partition of the path (s j+2 , s j+3 , . . . , s mk−i−1 ), i.e. a path on mk − (i + 1) − (j + 1) = (m − 1)k vertices.  ... 
doi:10.1016/j.endm.2018.05.009 fatcat:6bdpnd5nvnhtdgoapkelxslvyu

Page 69 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
(CH-LSNP) Paths, chains, and antipaths. Networks 19 (1989), no. 1, 107-115. An Eulerian partition of a multigraph G = (V, E) is a partition of E into a minimum number of chains.  ...  The paper relates this concept to that of the toughness of a graph G with an even number p of vertices.  ... 

Powers of Hamiltonian paths in interval graphs

Garth Isaak
1998 Journal of Graph Theory  
We give a simple proof that the obvious necessary conditions for a graph to contain the k th power of a Hamiltonian path are sufficient for the class of interval graphs.  ...  The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs.  ...  ''Connecting'' two parts of a minimum size path power partition into a single path power requires at least one extra vertex and at most k extra vertices.  ... 
doi:10.1002/(sici)1097-0118(199805)28:1<31::aid-jgt3>3.0.co;2-g fatcat:c457kvbi2fhv7kfqaiufzjwidu

Powers of Hamiltonian paths in interval graphs

Garth Isaak
1998 Journal of Graph Theory  
We give a simple proof that the obvious necessary conditions for a graph to contain the k th power of a Hamiltonian path are sufficient for the class of interval graphs.  ...  The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs.  ...  ''Connecting'' two parts of a minimum size path power partition into a single path power requires at least one extra vertex and at most k extra vertices.  ... 
doi:10.1002/(sici)1097-0118(199805)28:1<31::aid-jgt3>3.3.co;2-w fatcat:lspinavnybe4jbjastn4c3ziiq

Forbidden subgraphs for hamiltonicity of 1-tough graphs

Hajo J. Broersma, Binlong Li, Shenggui Zhang
2016 Discussiones Mathematicae Graph Theory  
A graph G is said to be 1-tough if for every vertex cut S of G, the number of components of G − S does not exceed |S|.  ...  We study the problem of characterizing all graphs H such that every 1-tough H-free graph is hamiltonian.  ...  We thank the anonymous referees for suggestions that improved the presentation of the paper.  ... 
doi:10.7151/dmgt.1897 fatcat:dmfmczvy4fdi5i56xrwuphu5ke

Page 5480 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Summary: “We characterize tough-maximum graphs, that is graphs having maximum number of edges among all graphs with given number of vertices and toughness.” 2002h:05093 05C35 Jiang, Zhi Ming (PRC-EUST;  ...  The results were obtained by considering a graphical function that measures the density of edges in subgraphs obtained by partitioning the vertices of a graph into two sets.  ... 

Toughness, degrees and 2-factors

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson, Linda Lesniak, Akira Saito
2004 Discrete Mathematics  
In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with (G) |V (G)|−4 2 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors.  ...  In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that 1 k n−16 4 .  ...  If V (G) is partitioned into sets S 1 , . . . , S k and the graph induced by each S i , denoted S i , contains a spanning cycle, we say that V (G) is partitioned into cycles C 1 , . . . , C k .  ... 
doi:10.1016/j.disc.2004.05.008 fatcat:yzk2l4r77jdqdgz3utr2xwai7e

Page 723 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
Given a complete graph of odd order, say Ky), a set of 2k —1 Eulerian tours which partition the set of 2-paths in Kx, is called a perfect set of Eulerian tours of the graph.  ...  An Eulerian tour of a graph G can be completely described by a list of the 2-paths (paths on 3 vertices) which the tour contains.  ... 

Page 7310 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
Let t(G) denote the number of vertices in a longest path P of the graph G.  ...  (RS-AOSSI; Novosibirsk) On path kernels and partitions in nondirected graphs. (Russian. Russian summary) Diskretn. Anal. Issled. Oper. Ser. 1 9 (2002), no. 2, 21-35.  ... 

Hamiltonian cycles in 7-tough (P_3∪ 2P_1)-free graphs [article]

Yuping Gao, Songling Shan
2021 arXiv   pre-print
The toughness of a noncomplete graph G is the maximum real number t such that the ratio of |S| to the number of components of G-S is at least t for every cutset S of G, and the toughness of a complete  ...  In this paper, we confirm Chvátal's toughness conjecture for (P_3∪ 2P_1)-free graphs by showing that every 7-tough (P_3∪ 2P_1)-free graph on at least three vertices is hamiltonian.  ...  Figure 1 : 1 1-tough nonhamiltonian graph with seven vertices Lemma 2. 1 . 1 ([1], Theorem 2.10) Let G be a bipartite graph with partite sets X and Y , and f be a function from X to the set of positive  ... 
arXiv:2107.08476v1 fatcat:3dyqqeeeajh4zll7dqub5kqtiy

Generalizating path and fan graphs: subcoloring and toughness

Lilian Markenzon, Christina F.E.M. Waga
2014 Pesquisa Operacional  
It is shown that the elements of the new classes establish bounds for the toughness of k-path graphs.  ...  Two graph classes are presented; the first one (k-ribbon) generalizes the path graph and the second one (k-fan) generalizes the fan graph.  ...  K 1 + R 4 K 2 + R 3 K 3 + R 2 K 4 + R 1 SUBCOLORING A s-coloring of a graph G = (V , E) is a partition of the vertices into s pairwise disjoint sets V 1 , . . . , V s such that for every i = 1, . . .  ... 
doi:10.1590/s0101-74382014005000004 fatcat:4n6t6z6q2jfv3erycso66jlcv4

Page 55 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
The final result is that for any set of points V and any partition of those points into k subsets, there exists a graph G with those k subsets forming a point-set domatic partition of G. Anthony E.  ...  A partition of the vertices of G is point-set domatic if all of its classes are point-set dominating in G.  ... 
« Previous Showing results 1 — 15 out of 2,298 results