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Vertex Colouring and Forbidden Subgraphs ? A Survey

Bert Randerath, Ingo Schiermeyer
2004 Graphs and Combinatorics  
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doi:10.1007/s00373-003-0540-1 fatcat:lmjw4ixws5fo7fjgmxl34t6wzu

Reconfiguration of vertex colouring and forbidden induced subgraphs [article]

Manoj Belavadi, Kathie Cameron, Owen Merkel
2022 arXiv   pre-print
In this paper, we investigate the connectivity and diameter of ℛ_k+1(G) for a k-colourable graph G restricted by forbidden induced subgraphs.  ...  The reconfiguration graph of the k-colourings, denoted ℛ_k(G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in ℛ_k(G) if they differ in colour on exactly one vertex  ...  The connectivity and diameter of R ℓ (G) has also been investigated for classes of graphs defined by more than one forbidden induced subgraph.  ... 
arXiv:2206.09268v2 fatcat:xsxdkirhd5gilk7pj4z7m3soou

A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs [article]

Petr A. Golovach, Matthew Johnson, Daniël Paulusma, Jian Song
2016 arXiv   pre-print
We survey known results on the computational complexity of Colouring and k-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs.  ...  We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.  ...  We thank Konrad Dabrowski, François Dross, Oliver Schaudt and two anonymous reviewers for helpful comments.  ... 
arXiv:1407.1482v8 fatcat:bjkwvdxmszgmnotdrv53epjt54

A survey of hereditary properties of graphs

Mieczysław Borowiecki, Izak Broere, Marietjie Frick, Peter Mihók, Gabriel Semanišin
1997 Discussiones Mathematicae Graph Theory  
In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs.  ...  The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized.  ...  The Choice Number Let G be a graph and let L(v) be a list of colours (say, positive integers) prescribed for the vertex v, and P ∈ M .  ... 
doi:10.7151/dmgt.1037 fatcat:qpxm2z2i6rhwra7lk42p2lg2au

A survey of χ-boundedness [article]

Alex Scott, Paul Seymour
2020 arXiv   pre-print
If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs?  ...  This is a survey of where we are now.  ...  We would particularly like to thank Pierre Aboulker, Pierre Charbit, Louis Esperet, Patrice Ossona de Mendez, Sang-Il Oum, Bruce Reed, Vaidy Sivaraman, Sophie Spirkl and Nicolas Trotignon.  ... 
arXiv:1812.07500v3 fatcat:q3xjtgcmkfh7fdzq2gnidx7lri

Reflexive cacti: A survey

Bojana Mihailovic, Marija Rasajski, Zoran Stanic
2016 Applicable Analysis and Discrete Mathematics  
A graph is called reflexive if its second largest eigenvalue does not exceed 2. We survey the results on reflexive cacti obtained in the last two decades.  ...  We also discuss various patterns of appearing of Smith graphs as subgraphs of reflexive cacti.  ...  The authors are grateful to the Serbian Ministry of Education, Science and Technological Development for financial support through Projects No 174033 and 174012.  ... 
doi:10.2298/aadm160929022m fatcat:ivneduwgivavnirsjxiy5iybqy

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Petr A. Golovach, Matthew Johnson, Daniël Paulusma, Jian Song
2016 Journal of Graph Theory  
We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs.  ...  2016) 'A survey on the computational complexity of coloring graphs with forbidden subgraphs.', Journal of graph theory. .  ...  We thank Konrad Dabrowski, François Dross, Oliver Schaudt and two anonymous reviewers for helpful comments.  ... 
doi:10.1002/jgt.22028 fatcat:mv5j3zmgjjfkhij5inlnbxdxhy

Simplicial decompositions of graphs: a survey of applications

Reinhard Diestel
1989 Discrete Mathematics  
We survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory.  ...  of B, to a given k-colouring of S,, and hence to any given k-colouring of GI, := l_l*__ Bh; thus by induction, G can be k-coloured if every Bh can.  ... 
doi:10.1016/0012-365x(89)90084-8 fatcat:n7g4dykpxnahzogrtopyf2nitq

Reconfiguration of Colourings and Dominating Sets in Graphs: a Survey [article]

C.M. Mynhardt, S. Nasserasr
2020 arXiv   pre-print
We survey results concerning reconfigurations of colourings and dominating sets in graphs.  ...  The vertices of the k-colouring graph C_k(G) of a graph G correspond to the proper k-colourings of a graph G, with two k-colourings being adjacent whenever they differ in the colour of exactly one vertex  ...  ; • C 3 , C 4 , C 6 are the only cycles that are colouring graphs; • every tree is a subgraph of a colouring graph (thus there is no finite forbidden subgraph characterisation of colouring graphs).  ... 
arXiv:2003.05956v1 fatcat:b2akpiblpvg6zk3n55ydlcp64y

A survey on enhanced power graphs of finite groups

Xuanlong Ma, Andrei Kelarev, Yuqing Lin, Kaishun Wang
2022 Electronic Journal of Graph Theory and Applications  
We survey known results on enhanced power graphs of finite groups. Open problems, questions and suggestions for future work are also included.  ...  Forbidden subgraphs in the enhanced power graph This section contains the results on the forbidden subgraphs of enhanced power graphs.  ...  Section 5 is devoted to the forbidden subgraphs of enhanced power graphs.  ... 
doi:10.5614/ejgta.2022.10.1.6 fatcat:r6p6q6ckebc6dbpcvxiorct4ay

Generalizations of tournaments: A survey

J�rgen Bang-Jensen, Gregory Gutin
1998 Journal of Graph Theory  
We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure.  ...  We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles.  ...  In [81] a characterization was given in terms of intersection graphs and a characterization by forbidden induced subgraphs was described.  ... 
doi:10.1002/(sici)1097-0118(199808)28:4<171::aid-jgt1>;2-g fatcat:urwonpylavfwzelzhggoojzk74

Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey

G. Gutin
1995 Journal of Graph Theory  
We describe results (theorems and algorithms) on directed walks in semicomplete m-partite digraphs including some recent results concerning tournaments.  ...  A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph.  ...  In particular, they proved that every tournament whose arcs are coloured with two colours contains a vertex v such that for every other vertex w there exists a monochromatic (v, w)-path.  ... 
doi:10.1002/jgt.3190190405 fatcat:7cnx4wtlobhcnjizv3ry7lnszi

A survey on integral graphs

Krystyna Balinska, Dragos Cvetkovic, Zoran Radosavljevic, Slobodan Simic, Dragan Stevanovic
2002 Publikacija Elektrotehnickog fakulteta - serija matematika  
A graph whose spectrum consists entirely of integers is called an integral graph. We present a survey of results on integral graphs and on the corresponding proof techniques.  ...  Notice that any subgraph of H, together with the induced labelling, produces an induced subgraph of G.  ...  ,n/k , on n vertices and k colour classes of sizes n/k, is always integral; the eigenvalues are: n − n/k, 0 n−k , (−n/k) k−1 .  ... 
doi:10.2298/petf0213042b fatcat:kw5yo33ib5dmxp6oar6nplauie

On some intriguing problems in hamiltonian graph theory—a survey

H.J. Broersma
2002 Discrete Mathematics  
We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.  ...  Introduction As this survey paper is the outcome of an invited lecture at the eighth Workshop on Cycles and Colourings (Starà a Lesnà a, Slovakia, 1999), the presentation of results is motivated by open  ...  Most of these results involve su cient conditions in terms of degrees, neighborhoods, forbidden subgraphs or (local) connectivity.  ... 
doi:10.1016/s0012-365x(01)00325-9 fatcat:wcggfjuntbgrjmegqldzdz55hy

A Survey of Forbidden Configuration Results

Richard Anstee
2013 Electronic Journal of Combinatorics  
Or alternatively if $A$ is an $m\times (\hbox{forb}(m,F)+1)$ simple matrix then $A$ has a submatrix which is a row and column permutation of $F$. We call $F$ a forbidden configuration.  ...  Let $F$ be a $k\times \ell$ (0,1)-matrix. We say a (0,1)-matrix $A$ has $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$.  ...  The investigations into the extremal problem of the maximum number of edges in an n vertex graph with no subgraph H originated with Erdős and Stone [ES46] and Simonovits [ES66] and has a large and  ... 
doi:10.37236/2379 fatcat:wxikdxtrqfbqxbdlbpc2dypuyu
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