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Colouring Square-Free Graphs without Long Induced Paths
[article]
2018
arXiv
pre-print
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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
unpublished
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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]
2017
arXiv
pre-print
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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
2019
Pure Mathematics and Applications
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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]
2018
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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
2014
Discrete Mathematics & Theoretical Computer Science
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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]
2012
Lecture Notes in Computer Science
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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
2005
Theoretical Computer Science
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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
2003
Discrete Mathematics
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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
2002
Discrete Mathematics
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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]
2016
arXiv
pre-print
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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]
2017
Lecture Notes in Computer Science
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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]
2015
arXiv
pre-print
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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
Preserved Fulltext
The Internet Archive has digitized a microfilm copy of this work. It may be possible to borrow a copy for reading.
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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