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Unique list-colourability and the fixing chromatic number of graphs

Amir Daneshgar, Hossein Hajiabolhassan
2005 Discrete Applied Mathematics  
In this paper we introduce a chromatic parameter, called the fixing chromatic number, which is related to unique colourability of graphs, in the sense that it measures how one can embed the given graph  ...  G in G ∪ K t by adding edges between G and K t to make the whole graph uniquely t-colourable.  ...  Acknowledgements Both authors are grateful for the financial support of the Institute for Studies in Theoretical Physics and Mathematics (IPM). Also, they wish to thank the anonymous referees and R.  ... 
doi:10.1016/j.dam.2005.04.006 fatcat:rflbcrjys5cthomwkq4jzych2m

Page 6733 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
: Tehran On some parameters related to uniquely vertex-colourable graphs and defining sets.  ...  The first method considered is to build a k-chromatic uniquely vertex colourable graph from a k-chromatic graph G on GU Ky by adding a set of new edges between the two components.  ... 

The parameterised complexity of counting connected subgraphs and graph motifs

Mark Jerrum, Kitty Meeks
2015 Journal of computer and system sciences (Print)  
We then apply these results to a counting version of the Graph Motif problem.  ...  We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show  ...  Let G = (V, E) be a graph on n vertices that is k-coloured by some colouring f : V (G) → [k], and let H be a k-vertex graph of treewidth t that is k-coloured by some colouring π such that H is colourful  ... 
doi:10.1016/j.jcss.2014.11.015 fatcat:xyrkofhgdndtbkmshkjwgukyui

The role of the Axiom of Choice in proper and distinguishing colourings [article]

Marcin Stawiski
2021 arXiv   pre-print
We investigate the role of the Axiom of Choice in the existence of certain proper or distinguishing colourings in both vertex and edge variants with special emphasis on locally finite connected graphs.  ...  We also formulate few conditions about distinguishing and proper colourings which are equivalent to the Axiom of Choice.  ...  This also holds for similarly defined parameters of different colourings. To see that, let Φ be some property of colourings of Ω.  ... 
arXiv:2012.00141v2 fatcat:uzkdop37wzhzxf45j6zwi5bjjq

The Parameterised Complexity of Counting Connected Subgraphs and Graph Motifs [article]

Mark Jerrum, Kitty Meeks
2014 arXiv   pre-print
We then apply these results to a counting version of the Graph Motif problem.  ...  We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show  ...  In order to define this hierarchy, we need some more notions related to satisfiability problems.  ... 
arXiv:1308.1575v3 fatcat:fryb77k3gze65clacn6kgap4pq

Uniquely (m, k)τ-colourable graphs and k − τ-saturated graphs

Gerhard Benadé, Izak Broere, Betsie Jonck, Marietjie Frick
1996 Discrete Mathematics  
A graph H is uniquely (m, kf-colourable if z~(H) = m and there is only one partition of the vertex set of H which is an (m, k)~-colouring of H.  ...  For k = 1, the graphs obtained by taking the join of k -z-saturated graphs (which are empty graphs in this case) are known to be uniquely colourable graphs.  ...  Following [6] , a graph H is called uniquely (m, k)~-colourable if it is (m, k)~-chromatic and there is only one partition of the vertex set of H which is an (m, k)~-colouring of H.  ... 
doi:10.1016/0012-365x(95)00301-c fatcat:atxykwtqlbftxbudemnm4rofwu

The b-chromatic number of a graph

Robert W. Irving, David F. Manlove
1999 Discrete Applied Mathematics  
Here we show that $(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G.  ...  We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the h-chmmtic number, q(G). of G.  ...  Acknowledgements The authors are grateful to the referees for their valuable comments. References [I] H.L. Bodlaender, Achromatic number is NP-complete for cographs and interval graphs, Inform.  ... 
doi:10.1016/s0166-218x(98)00146-2 fatcat:wxkmye3pvnaxre4n6f5iykntza

On defining numbers of vertex colouring of regular graphs

E.S. Mahmoodian, E. Mendelsohn
1999 Discrete Mathematics  
In a given graph G, a set S of vertices with an assignment of colours to them is a defining set of the vertex colourin 9 of G, if there exists a unique extension of the colours of S to a z(G)-colouring  ...  A defining set with minimum cardinality is called a minimum defining set (of vertex colouring) and its cardinality, the defining number, is denoted by d(G, Z).  ...  Acknowledgements Part of the research of the first author was done in the Department of Mathematics of the University of Toronto, Canada while he was on sabbatical leave.  ... 
doi:10.1016/s0012-365x(99)90113-9 fatcat:heiaxbnkpfaszjkyssp4xoueam

Page 1336 of Mathematical Reviews Vol. , Issue 85d [page]

1985 Mathematical Reviews  
edge-colouring of some semiregular graphs.  ...  This is true, for example, of the set of graphs homeomorphic to K4, the complete graph on four vertices.  ... 

Automorphism Properties and Classification of Adinkras

B. L. Douglas, S. James Gates, B. L. Segler, J. B. Wang
2015 Advances in Mathematical Physics  
In this paper we efficiently classify the automorphism groups of Adinkras relative to a set of local parameters.  ...  Using this, we classify Adinkras according to their equivalence and isomorphism classes.  ...  Acknowledgment The authors would like to thank Akihiro Munemasa for detailed comments and suggestions on this paper, as well as Cheryl Praeger, Cai-Heng Li, Gordon Royle, and John Bamberg for several useful  ... 
doi:10.1155/2015/584542 fatcat:5o7avx262necnaqxsuqvvk6i64

Conflict-Free Colourings of Graphs and Hypergraphs

JÁNOS PACH, GÁBOR TARDOS
2009 Combinatorics, probability & computing  
For example, it is shown that the vertices of any graph G with maximum degree Δ can be coloured with log 2+ Δ colours, so that the neighbourhood of every vertex contains a point of 'unique' colour.  ...  This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks.  ...  Conflict-Free Colourings of Graphs and Hypergraphs  ... 
doi:10.1017/s0963548309990290 fatcat:47cm2tiqqvefdp4jdq7phjjx3q

Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion

Jan van den Heuvel, Stephan Kreutzer, Michal Pilipczuk, Daniel A. Quiroz, Roman Rabinovich, Sebastian Siebertz
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
with some (and hence every) successor relation is a model of a given formula.  ...  Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.  ...  Again, this is an area where further research seems appropriate. is the graph on the vertex set A in which two elements u, v ∈ A are adjacent if and only if u = v and u and v appear together in some relation  ... 
doi:10.1109/lics.2017.8005115 dblp:conf/lics/HeuvelKPQRS17 fatcat:jdqaqsza6rgi3ke7ldtaekaj6i

Tropical Dominating Sets in Vertex-Coloured Graphs [article]

J.-A. Angles d'Auriac, Cs. Bujtas, A. El Maftouhi, M. Karpinski, Y. Manoussakis, L. Montero, N. Narayanan, L. Rosaz, J. Thapper, Zs. Tuza
2016 arXiv   pre-print
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set.  ...  Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give upper bounds for random graphs.  ...  Bujtás and Zs. Tuza was supported in part by the European Union and Hungary, co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004.  ... 
arXiv:1503.01008v3 fatcat:lgim7b5iujcivmzdsfaqukb3bq

Model-Checking for Successor-Invariant First-Order Formulas on Graph Classes of Bounded Expansion [article]

Jan van den Heuvel and Stephan Kreutzer and Michał Pilipczuk and Daniel A. Quiroz and Roman Rabinovich and Sebastian Siebertz
2017 arXiv   pre-print
with some (and hence every) successor relation is a model of a given formula.  ...  Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic.  ...  Again, this is an area where further research seems appropriate. is the graph on the vertex set A in which two elements u, v ∈ A are adjacent if and only if u = v and u and v appear together in some relation  ... 
arXiv:1701.08516v2 fatcat:pmoy3mbenjchxa6xr6sbapy72y

Tropical dominating sets in vertex-coloured graphs

J.-A. Anglès d'Auriac, Cs. Bujtás, A. El Maftouhi, M. Karpinski, Y. Manoussakis, L. Montero, N. Narayanan, L. Rosaz, J. Thapper, Zs. Tuza
2018 Journal of Discrete Algorithms  
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set.  ...  Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give an optimal upper bound for random graphs.  ...  Bujtás and Zs. Tuza was supported in part by the European Union and Hungary, co-financed by the European Social Fund through the project TÁMOP-4.2.2.C-11/1/KONV-2012-0004.  ... 
doi:10.1016/j.jda.2018.03.001 fatcat:ryey47vaajemvk573j5btpsram
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