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### Page 1329 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews
(D-MUTU; Munich) Hamiltonicity in graphs with few P,’s. (English and German summaries) Computing 54 (1995), no. 3, 213-225.  ...  05C45 Ryja¢éek, Zdenék (CZ-WSTB; Pizen) Hamiltonicity in claw-free graphs through induced bulls.  ...

### Hamilton cycles in random subgraphs of pseudo-random graphs

Alan Frieze, Michael Krivelevich
2002 Discrete Mathematics
Given an r-regular graph G on n vertices with a Hamilton cycle, order its edges randomly and insert them one by one according to the chosen order, starting from the empty graph.  ...  As a consequence we derive the value of the threshold for the appearance of a Hamilton cycle in a random subgraph of a pseudo-random graph G, satisfying the above stated condition. (A.  ...  Thus, the corresponding graph process problem can be formulated in general as follows: given a graph G with a Hamilton cycle, is it true that for almost all graph processes (G m ) the ÿrst Hamilton cycle  ...

### Optimal covers with Hamilton cycles in random graphs

Dan Hefetz, Daniela Kühn, John Lapinskas, Deryk Osthus
2014 Combinatorica
A packing of a graph G with Hamilton cycles is a set of edgedisjoint Hamilton cycles in G.  ...  Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.  ...  In that theorem, given a set S of vertices in a graph G, we let N (S) denote the external neighbourhood of S, i.e. the set of all those vertices x / ∈ S for which there is some vertex y ∈ S with xy ∈ E  ...

### Optimal covers with Hamilton cycles in random graphs [article]

Dan Hefetz and Daniela Kühn and John Lapinskas and Deryk Osthus
2013 arXiv   pre-print
A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G.  ...  Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.  ...  In that theorem, given a set S of vertices in a graph G, we let N (S) denote the external neighbourhood of S, i.e. the set of all those vertices x / ∈ S for which there is some vertex y ∈ S with xy ∈ E  ...

### Triangulating graphs with few P4's

Luitpold Babel
1998 Discrete Applied Mathematics
As a direct application we show how to compute in linear time the minimum fill-in and the treewidth of graphs whichin a certain local sensecontain only a small number of induced P4's.  ...  In this paper we point out the use of the homogeneous decomposition of graphs for the purpose of triangulation.  ...  This comes into line with other problems that can be solved in linear time on graphs with few Pd's, e.g. maximum clique, minimum stable set, minimum coloring [3, 201, hamiltonicity [ 10, 161, dominating  ...

### On a unique tree representation for P4-extendible graphs

B. Jamison, S. Olariu
1991 Discrete Applied Mathematics
Jamison, B. and S. Olariu, On a unique tree representation for P4-extendible graphs, Discrete Applied Mathematics 34 (1991) 151-164.  ...  Several practical applications in computer science and computational linguistics suggest the study of graphs that are unlikely to have more than a few induced paths of length three.  ...  Several practical applications in computer science and computational linguistics suggest the study of graphs that are unlikely to have more than a few induced P4's.  ...

### Random graph's Hamiltonicity is strongly tied to its minimum degree [article]

Yahav Alon, Michael Krivelevich
2019 arXiv   pre-print
We show that the probability that a random graph G∼ G(n,p) contains no Hamilton cycle is (1+o(1))Pr(δ (G) < 2) for all values of p = p(n). We also prove an analogous result for perfect matchings.  ...  Recall that in the proof of Lemma 3.8 we showed that in a graph with properties (R1)-(R3), removing a minimum degree vertex yields a Hamilton-connected graph.  ...  of Hamilton cycles in random graphs obtained over recent years.  ...

### Author index to volume 191 (1998)

1998 Discrete Mathematics
Stiebitz, Colour-critical graphs with few edges (1--3} 125 137 Kratochvil, J., Zs. Tuza and M.  ...  Schelp, Vertex colorings with a distance restriction (1-3) 65 82 Fabrici, I. and S.  ...

### Page 8601 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews
with few P,4’s (29-51); S.  ...  Bylka, Local improving algorithms for large cuts in graphs with maximum degree three (53-67); Kathie Cameron, Thomason’s algorithm for finding a second Hamilton- ian circuit through a given edge in a cubic  ...

### Finding Hamilton cycles in random intersection graphs [article]

Katarzyna Rybarczyk
2017 arXiv   pre-print
In this article we study the problem of finding a Hamilton cycle in a random intersection graph.  ...  To this end we analyse a classical algorithm for finding Hamilton cycles in random graphs (algorithm HAM) and study its efficiency on graphs from a family of random intersection graphs (denoted here by  ...  Main results Algorithm HAM, as presented in  , is designed to search for a Hamilton cycle in any graph with the minimum degree at least 2.  ...

### Stability for Vertex Cycle Covers

József Balogh, Frank Mousset, Jozef Skokan
2017 Electronic Journal of Combinatorics
$cycles in$G$that together cover all the vertices.This is tight in the sense that there are$n$-vertex graphs that have minimum degree$n/k-1$and that do not contain$k-1$cycles with this property ... In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer$k\geq 2$, if$G$is an$n$-vertex graph with minimum degree at least$n/k$, then there are$k-1  ...  Finally, we will absorb the few remaining vertices of G i into the path to get a Hamilton path.  ...

### Colour-biased Hamilton cycles in random graphs [article]

Lior Gishboliner, Michael Krivelevich, Peleg Michaeli
2021 arXiv   pre-print
We prove that a random graph G(n,p), with p above the Hamiltonicity threshold, is typically such that for any r-colouring of its edges there exists a Hamilton cycle with at least (2/(r+ 1)-o(1))n edges  ...  It is natural to expect that if G contains only few Hamilton cycles, then one can r-colour the edges of G in such a way that every Hamilton cycle sees approximately the same number, i.e. roughly n/r, of  ...  Write s = 1/ε , let t = n/s ∼ ε n, and let A 1 , . . . , A t , Z be a partitioning of the vertices of G into t "blobs" A i of size s and an extra set Z with |Z| ≤ s.  ...

### Random Graph's Hamiltonicity is Strongly Tied to its Minimum Degree

Yahav Alon, Michael Krivelevich
2020 Electronic Journal of Combinatorics
We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$.  ...  Recall that in the proof of Lemma 21 we showed that in a graph with properties (R1)-(R3), removing a minimum degree vertex yields a Hamilton-connected graph.  ...  A graph G is Hamilton-connected if for every two vertices u, v ∈ V (G), G contains a Hamilton path with u, v as its two endpoints. Theorem 7.  ...

### Resilience of Perfect Matchings and Hamiltonicity in Random Graph Processes [article]

Rajko Nenadov, Angelika Steger, Miloš Trujić
2018 arXiv   pre-print
Let {G_i} be the random graph process: starting with an empty graph G_0 with n vertices, in every step i ≥ 1 the graph G_i is formed by taking an edge chosen uniformly at random among the non-existing  ...  In particular, we show that the random graph process almost surely creates a sequence of graphs such that for m ≥ (16 + o(1))n n edges, the 2-core of the graph G_m remains Hamiltonian even after an adversary  ...  The third author would like to thank Michael Krivelevich for directing his attention to  which helped in making some of the arguments cleaner.  ...

### Master Index: Volumes 81–902-90

1999 Discrete Applied Mathematics
Olariu, On the structure of graphs with few P& 84 (1998) 1m 13 Babel, L., Triangulating graphs with few P4's 89 (1998) 45m 57 Balas. E. and M.  ...  -H., The exact bound of Lee's MLPT Lin, R. and S. Olariu, A fast parallel algorithm to recognize P4-sparse graphs Lippens, P.E.R., see W.F.J. Verhaegh Litman, A., see S. Ben-David Litsyn, S., P.  ...
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