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Unique list-colourability and the fixing chromatic number of graphs
2005
Discrete Applied Mathematics
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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
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: 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
2015
Journal of computer and system sciences (Print)
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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]
2021
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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
1996
Discrete Mathematics
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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
1999
Discrete Applied Mathematics
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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
1999
Discrete Mathematics
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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
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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
2015
Advances in Mathematical Physics
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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
2009
Combinatorics, probability & computing
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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
2017
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
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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]
2016
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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
2018
Journal of Discrete Algorithms
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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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