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Colouring Square-Free Graphs without Long Induced Paths
[article]

2018
*
arXiv
*
pre-print

The complexity of

arXiv:1805.08270v1
fatcat:alqgbles2ncihj3yahfe2w5zc4
*Colouring*is fully understood for H-*free**graphs*, but there are still major complexity gaps if two*induced*subgraphs H_1 and H_2 are forbidden. ... As a complementary result we prove that*Colouring*is -complete for s=4 and t≥ 9, which is the first hardness result on*Colouring*for (C_4,P_t)-*free**graphs*. ... Initially we proved NP-hardness of*Colouring*for (C 4 , P 16 )-*free**graphs*. Afterwards we were able to improve this result to (C 4 , P 9 )-*free**graphs*via a simplification of our construction. ...##
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Colouring Square-Free Graphs without Long Induced Paths

unpublished

Our algorithm is based on a novel decomposition theorem for (C 4 , P 6)-

fatcat:iyvsws4otbhtvg2krh4ukvna3u
*free**graphs**without*clique cutsets into homogeneous pairs of sets and a new framework for bounding the clique-width of a*graph*by ... Acknowledgements Initially we proved NP-hardness of*Colouring*for (C 4 , P 16)-*free**graphs*. ... Acknowledgements Initially we proved NP-hardness of*Colouring*for (C 4 , P 16 )-*free**graphs*. ...##
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Anagram-free colourings of graph subdivisions
[article]

2017
*
arXiv
*
pre-print

A vertex

arXiv:1708.09571v1
fatcat:x57teqhvwzbwtjkpdomescgqbi
*colouring*of a*graph*is anagram-*free*if no subpath of the*graph*is an anagram. Anagram-*free**graph**colouring*was independently introduced by Kamčev, Łuczak and Sudakov and ourselves. ... In this paper we introduce the study of anagram-*free**colourings*of*graph*subdivisions. We show that every*graph*has an anagram-*free*8-*colourable*subdivision. ... Let u be the other endpoint of α, P α be the*path**induced*by the division vertices of α, and P β be the*path**induced*by the division vertices of β. Then uP α vP β is an anagram in S. ...##
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Colourability and word-representability of near-triangulations

2019
*
Pure Mathematics and Applications
*

It is known [6] that all 3-

doi:10.1515/puma-2015-0032
fatcat:sgowidk73jegjoiahvu77cprre
*colourable**graphs*are word-representable, while among those with a higher chromatic number some are word-representable while others are not. ... A*graph*G = (V;E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x; y) is in G. ... Lemma 2.2 Let x be a vertex of odd degree in a*graph*G, and suppose the neighbours of x*induce*a*path*P . In any 3-*colouring*of G, the end-points of P must have the same*colour*. Proof. ...##
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Colourability and word-representability of near-triangulations
[article]

2018
*
arXiv
*
pre-print

It is known [6] that all 3-

arXiv:1605.01688v4
fatcat:xarieqfk2ngrrnvxvgdyhws6xq
*colourable**graphs*are word-representable, while among those with a higher chromatic number some are word-representable while others are not. ... A*graph*G = (V,E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x, y) is in G. ... A K 4 -*free*near-triangulation is 3-*colourable*if and only if it is word-representable. Lemma 1. Let x be a vertex of odd degree in a*graph*G, and suppose the neighbours of x*induce*a*path*P . ...##
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Anagram-free Graph Colouring
[article]

2017
*
arXiv
*
pre-print

We study anagram-

arXiv:1607.01117v2
fatcat:pwrf7dcidjaylex5bqydehs36u
*free**graph**colouring*and give bounds on the chromatic number. Alon et al. (2002) asked whether anagram-*free*chromatic number is bounded by a function of the maximum degree. ... Finally, we explore extensions to edge*colouring*and k-anagram-*free**colouring*. ... Note At the same time as the present paper was completed, Kamčev, Łuczak, and Sudakov [15] posted a paper on the arXiv that independently introduces anagram-*free**graph**colouring*. ...##
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Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

2014
*
Discrete Mathematics & Theoretical Computer Science
*

*graphs*and bipartite

*graphs*containing no 4-cycles as

*induced*subgraphs. ...

*Graph*Theory International audience Given a

*graph*and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the

*graph*can be partitioned into at most k bicliques (connected ... Partitioning the vertex set of a bipartite

*graph*into bicliques ...

##
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On the Complexity of Rolling Block and Alice Mazes
[chapter]

2012
*
Lecture Notes in Computer Science
*

The question of the complexity with only 2 × 1 × 1 blocks, and

doi:10.1007/978-3-642-30347-0_22
fatcat:d3neevn4jvb2zlilj42gst77aa
*without*forbidden*squares*was left open-see also [9] . ... A slightly different variant are*colour*mazes, where the fields of playing board are*coloured*, and the block may only come to lie on a uni-*coloured*area. ... Since all*paths*, the block should roll along, are embedded by this alternating*colour*structure, the block cannot leave the*path*from an upright position, for it needs two uni-*coloured**squares*to lie on ...##
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Pattern avoidance: themes and variations

2005
*
Theoretical Computer Science
*

For each

doi:10.1016/j.tcs.2005.01.004
fatcat:cr5bmro63zbbnco3ogmsuw7yly
*induced**path*Q in P, let A Q be the event that the*colouring*of Q corresponds to a word in L. ... The problem of*square*-*free*walks on general*graphs*is attacked in [2] , where it is shown that a (possibly infinite)*graph*of bounded degree has a*colouring*with finitely many*colours*such that every ...##
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Finding a maximum induced matching in weakly chordal graphs

2003
*
Discrete Mathematics
*

It is known that ÿnding an

doi:10.1016/s0012-365x(02)00803-8
fatcat:rlvvp22rrrh4xoxmlh7zdrs5zu
*induced*matching of maximum cardinality in a*graph*is NP-hard. We show that a maximum*induced*matching in a weakly chordal*graph*can be found in polynomial time. ... An*induced*matching in a*graph*G is a set of edges, no two of which meet a common vertex or are joined by an edge of G; that is, an*induced*matching is a matching which forms an*induced*subgraph. ... In the following lemma, we prove that if G had no*long**induced*cycles, then G * does not have*long**induced*cycles either. Lemma 2. ...##
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Fibrations of graphs

2002
*
Discrete Mathematics
*

A ÿbration of

doi:10.1016/s0012-365x(00)00455-6
fatcat:56ylwecaybeydnryzlycqiczam
*graphs*is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. ... This paper develops systematically the theory of*graph*ÿbrations, emphasizing in particular those results that recently found application in the theory of distributed systems. ... Moreover, it is naturally*coloured*on the set S -the s-*induced*arc from Hg to Hgs is*coloured*by s; such a*graph*is called the Schreier*colour**graph*of with respect to H and S. ...##
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Improved Lower Bounds for Graph Embedding Problems
[article]

2016
*
arXiv
*
pre-print

do not exist algorithms that run in 2^o(n/ n) time for Subgraph Isomorphism on

arXiv:1610.09130v1
fatcat:tzn3x2czc5dfpiunxetynbd6ii
*graphs*of pathwidth 1,*Induced*Subgraph Isomorphism on*graphs*of pathwidth 1,*Graph*Minor on*graphs*of pathwidth 1,*Induced*...*Graph*Minor on*graphs*of pathwidth 1, Intervalizing 5-Colored*Graphs*on trees, and finding a tree or*path*decomposition with width at most c with a minimum number of bags, for any fixed c≥ 16. 2^Θ(n/ ... for Scientific Research NWO. and (*Induced*) Minor and is in fact tight: the problems admit 2 O(n/ log n) -time algorithms on H-minor*free**graphs*[3] . ...##
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Improved Lower Bounds for Graph Embedding Problems
[chapter]

2017
*
Lecture Notes in Computer Science
*

do not exist algorithms that run in 2 o(n/ log n) time for Subgraph Isomorphism on

doi:10.1007/978-3-319-57586-5_9
fatcat:5quyvsckf5bs3cbe3ugmqcv7fy
*graphs*of pathwidth 1,*Induced*Subgraph Isomorphism on*graphs*of pathwidth 1,*Graph*Minor on*graphs*of pathwidth 1,*Induced*...*Graph*Minor on*graphs*of pathwidth 1, Intervalizing 5-Colored*Graphs*on trees, and finding a tree or*path*decomposition with width at most c with a minimum number of bags, for any fixed c ≥ 16. 2 Θ(n/ ... for Scientific Research NWO. and (*Induced*) Minor and is in fact tight: the problems admit 2 O(n/ log n) -time algorithms on H-minor*free**graphs*[3] . ...##
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Rainbow matchings and rainbow connectedness
[article]

2015
*
arXiv
*
pre-print

Our proofs involve studying connectedness in

arXiv:1504.05373v1
fatcat:2dtdzfdkgbed3fmiaaxctot6ji
*coloured*, directed*graphs*. The notion of connectedness that we introduce is new, and perhaps of independent interest. ... This conjecture is a generalization of several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin*squares*. ... Also notice that it is possible for a subset A ⊆ V (D) to be highly connected*without*the*induced*subgraph D[A] being highly connected-indeed if D is a bipartite*graph*with classes X and Y where all edges ...##
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Page 3088 of Mathematical Reviews Vol. , Issue 94f
[page]

1994
*
Mathematical Reviews
*

In particular the Hamming

*graphs*are characterized as the interval- regular*graphs**without**induced*Kj. and isometric odd cycles other than triangles.” ... The corresponding extremal*graphs*can be determined at the same time, and sufficient information on Ce-*free*3-*colourings*is obtained to show that the Ramsey number r3(Cg) is equal to 12.” 94f:05108 05C65 ...
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