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

Serge Gaspers and Shenwei Huang and Daniël Paulusma
2018 arXiv   pre-print
The complexity of 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.  ... 
arXiv:1805.08270v1 fatcat:alqgbles2ncihj3yahfe2w5zc4

Colouring Square-Free Graphs without Long Induced Paths

Serge Gaspers, Shenwei Huang, Daniël Paulusma
unpublished
Our algorithm is based on a novel decomposition theorem for (C 4 , P 6)-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.  ... 
fatcat:iyvsws4otbhtvg2krh4ukvna3u

Anagram-free colourings of graph subdivisions [article]

Tim E. Wilson, David R. Wood
2017 arXiv   pre-print
A vertex 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.  ... 
arXiv:1708.09571v1 fatcat:x57teqhvwzbwtjkpdomescgqbi

Colourability and word-representability of near-triangulations

Marc Elliot Glen
2019 Pure Mathematics and Applications  
It is known [6] that all 3-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.  ... 
doi:10.1515/puma-2015-0032 fatcat:sgowidk73jegjoiahvu77cprre

Colourability and word-representability of near-triangulations [article]

Marc Elliot Glen
2018 arXiv   pre-print
It is known [6] that all 3-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 .  ... 
arXiv:1605.01688v4 fatcat:xarieqfk2ngrrnvxvgdyhws6xq

Anagram-free Graph Colouring [article]

Tim E. Wilson, David R. Wood
2017 arXiv   pre-print
We study anagram-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.  ... 
arXiv:1607.01117v2 fatcat:pwrf7dcidjaylex5bqydehs36u

Partitioning the vertex set of a bipartite graph into complete bipartite subgraphs

Oleg Duginov
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  ... 
doi:10.46298/dmtcs.2090 fatcat:nctfdlgk7vazlogbq3buoqz3gu

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

Markus Holzer, Sebastian Jakobi
2012 Lecture Notes in Computer Science  
The question of the complexity with only 2 × 1 × 1 blocks, and 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  ... 
doi:10.1007/978-3-642-30347-0_22 fatcat:d3neevn4jvb2zlilj42gst77aa

Pattern avoidance: themes and variations

James D. Currie
2005 Theoretical Computer Science  
For each 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  ... 
doi:10.1016/j.tcs.2005.01.004 fatcat:cr5bmro63zbbnco3ogmsuw7yly

Finding a maximum induced matching in weakly chordal graphs

Kathie Cameron, R. Sritharan, Yingwen Tang
2003 Discrete Mathematics  
It is known that ÿnding an 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.  ... 
doi:10.1016/s0012-365x(02)00803-8 fatcat:rlvvp22rrrh4xoxmlh7zdrs5zu

Fibrations of graphs

Paolo Boldi, Sebastiano Vigna
2002 Discrete Mathematics  
A ÿbration of 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.  ... 
doi:10.1016/s0012-365x(00)00455-6 fatcat:56ylwecaybeydnryzlycqiczam

Improved Lower Bounds for Graph Embedding Problems [article]

Hans L. Bodlaender, Tom C. van der Zanden
2016 arXiv   pre-print
do not exist algorithms that run in 2^o(n/ n) time for Subgraph Isomorphism on 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] .  ... 
arXiv:1610.09130v1 fatcat:tzn3x2czc5dfpiunxetynbd6ii

Improved Lower Bounds for Graph Embedding Problems [chapter]

Hans L. Bodlaender, Tom C. van der Zanden
2017 Lecture Notes in Computer Science  
do not exist algorithms that run in 2 o(n/ log n) time for Subgraph Isomorphism on 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] .  ... 
doi:10.1007/978-3-319-57586-5_9 fatcat:5quyvsckf5bs3cbe3ugmqcv7fy

Rainbow matchings and rainbow connectedness [article]

Alexey Pokrovskiy
2015 arXiv   pre-print
Our proofs involve studying connectedness in 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  ... 
arXiv:1504.05373v1 fatcat:2dtdzfdkgbed3fmiaaxctot6ji

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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