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Colorful induced subgraphs

H.A. Kierstead, W.T. Trotter
1992 Discrete Mathematics  
Colorful induced subgraphs, Discrete Mathematics 101 (1992) 165-169. A colored graph is a graph whose vertices have been properly, though not necessarily optimally colored, with integers.  ...  A subgraph of a colored graph is colorful if each of its vertices has a distinct color.  ...  The colored subgraph (subdigraph) of G induced by H is G[H] = (V, E', f '), where (V, E') is the subgraph (subdigraph) of (V, E) induced by H and f' is f restricted to H.  ... 
doi:10.1016/0012-365x(92)90600-k fatcat:cqqir6eq2bhkbn533na4ualy3u

Rainbow induced subgraphs in proper vertex colorings [article]

Andrzej Kisielewicz, Marek Szykuła
2011 arXiv   pre-print
For a given graph H we define ρ(H) to be the minimum order of a graph G such that every proper vertex coloring of G contains a rainbow induced subgraph isomorphic to H.  ...  This research is motivated by some ideas in on-line graph coloring algorithms.  ...  In this paper we deal with proper vertex colorings of graphs, colorings, in short, and rainbow induced subgraphs, by which we mean induced subgraphs whose all vertices have different colors.  ... 
arXiv:1105.3712v1 fatcat:ai2j75cgtndi5ghsoukxnu5zsu

Avoiding rainbow induced subgraphs in vertex-colorings [article]

Maria Axenovich, Ryan Martin
2016 arXiv   pre-print
For a fixed graph H on k vertices, and a graph G on at least k vertices, we write G→ H if in any vertex-coloring of G with k colors, there is an induced subgraph isomorphic to H whose vertices have distinct  ...  This determines the induced vertex-anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable.  ...  of R t (H) are colored with t colors, then there is an induced subgraph of R t (H) isomorphic to H which is monochromatic.  ... 
arXiv:1605.06172v1 fatcat:wqby2wraineb5n5dlnqlg65xfe

Complexity dichotomy for List-5-Coloring with a forbidden induced subgraph [article]

Sepehr Hajebi, Yanjia Li, Sophie Spirkl
2021 arXiv   pre-print
Also, we say G is H-free if H is not isomorphic to an induced subgraph of G. We use P_t to denote the path on t vertices.  ...  restricted to H-free graphs can be solved in polynomial time if and only if H is an induced subgraph of either rP_3 or P_5+rP_1 for some positive integer r.  ...  If H is an induced subgraph of rP 3 , then the result follows from Theorem 14, and if H is an induced subgraph of P 5 + rP 1 , then the result follows from Theorem 7.  ... 
arXiv:2105.01787v2 fatcat:be63sbjvhbfjjdhxt2mvk4el2y

Avoiding Rainbow Induced Subgraphs in Vertex-Colorings

Maria Axenovich, Ryan Martin
2008 Electronic Journal of Combinatorics  
For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\longrightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic  ...  This determines the induced vertex-anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable.  ...  of R t (H) are colored with t colors, then there is an induced subgraph of R t (H) isomorphic to H which is monochromatic.  ... 
doi:10.37236/736 fatcat:3lnbwdgxifgjjmbdrb7ef7ewwy

On the complexity of finding large odd induced subgraphs and odd colorings [article]

Rémy Belmonte, Ignasi Sau
2021 arXiv   pre-print
We study the complexity of the problems of finding, given a graph G, a largest induced subgraph of G with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition  ...  Now note that every proper vertex coloring of G using q colors can be lifted to a partition of V (G) into q odd induced subgraphs (in fact, odd induced forests).  ...  Tight bounds In this section we provide two tight bounds concerning odd induced subgraphs and odd colorings.  ... 
arXiv:2002.06078v2 fatcat:kqtwnbdf45h37fszgz2baxzlsm

Complexity of Coloring Graphs without Forbidden Induced Subgraphs [chapter]

Daniel Král', Jan Kratochvíl, Zsolt Tuza, Gerhard J. Woeginger
2001 Lecture Notes in Computer Science  
We further initiate a study of this problem for two forbidden subgraphs.  ...  We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete.  ...  If H is an induced subgraph of H then every H-free graph is also H -free, and hence H-Free Coloring ∝ H -Free Coloring.  ... 
doi:10.1007/3-540-45477-2_23 fatcat:bykjq7v4tjbg5jzuyaxlh7y6ly

Forbidden Induced Subgraph Of The Comparability Graph And Three Colored Posets

Baiju Sukumaran, Sibi C Babu, Athul T B
2018 Zenodo  
As a continuation of the study of 3-colored diagrams we characterize some forbidden ⊲ - preserving subposets of the posets whose cover-incomparability graph contains one of the forbidden induced subgraph  ...  We call posets having the above mentioned diagrams as 3-colored diagrams. Let F (G) be a collection of forbidden induced subgraphs.  ...  FIGURE 1 : 1 Families of forbidden induced subgraphs for comparability graph [14] Figure 2 : 2 Forbidden 3-colored diagrams for posets whose C-I graphs contains depicted inFigure 1(c).  ... 
doi:10.5281/zenodo.1412751 fatcat:4vkdjxz73jhoti2woxkarnpsym

Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs

Neeldhara Misra, Fahad Panolan, Ashutosh Rai, Venkatesh Raman, Saket Saurabh
2018 Algorithmica  
We address the parameterized complexity of Max Colorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G.  ...  -The second algorithm runs in time q +o( ) n O(1) Tα where Tα is the time required to find a maximum independent set in any induced subgraph of G.  ...  An interesting direction of research that this paper opens up is the study of parameterized complexity of Induced Subgraph Isomorphism on special graph classes.  ... 
doi:10.1007/s00453-018-0431-8 fatcat:2zlvhsjwozbyrjt3lezhobrdyy

Parameterized Algorithms for Max Colorable Induced Subgraph Problem on Perfect Graphs [chapter]

Neeldhara Misra, Fahad Panolan, Ashutosh Rai, Venkatesh Raman, Saket Saurabh
2013 Lecture Notes in Computer Science  
We address the parameterized complexity of Max Colorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G.  ...  -The second algorithm runs in time q +o( ) n O(1) Tα where Tα is the time required to find a maximum independent set in any induced subgraph of G.  ...  An interesting direction of research that this paper opens up is the study of parameterized complexity of Induced Subgraph Isomorphism on special graph classes.  ... 
doi:10.1007/978-3-642-45043-3_32 fatcat:tyxfpsbqm5bgdcrzzgdmjuiqea

The complexity of the 3-colorability problem in the absence of a pair of small forbidden induced subgraphs

D.S. Malyshev
2015 Discrete Mathematics  
We completely determine the complexity status of the 3-colorability problem for hereditary graph classes defined by two forbidden induced subgraphs with at most five vertices.  ...  If it exists, then G is 3-colorable if and only if N(x) induces a bipartite subgraph. Hence, we have a polynomial-time reduction.  ...  If there are at least two nonempty sets, then each of them induces a complete subgraph.  ... 
doi:10.1016/j.disc.2015.04.019 fatcat:5kklqcsohbauvg2c543a42bfve

Some perfect coloring properties of graphs

Claude A Christen, Stanley M Selkow
1979 Journal of combinatorial theory. Series B (Print)  
Notice that for positive integers n a graph G has a complete n-coloring if every finite induced subgraph H of G has one. Now let H be an arbitrary induced subgraph of G which satisfies (8).  ...  Hence, if G is yx-perfect, it does not contain an induced subgraph isomo.rphic to P4. (3) =+-(1) Suppose G does not contain an induced subgraph isomorphic to P4, and let c be a Grundy y(G)-coloring of  ... 
doi:10.1016/0095-8956(79)90067-4 fatcat:hvrz35tav5gxjj5giack7lud5a

A Characterization of 2-Tree Proper Interval 3-Graphs

David E. Brown, Breeann M. Flesch
2014 Journal of Discrete Mathematics  
We characterize the class of 2-trees which are interval 3-graphs via a list of three graphs and three infinite families of forbidden induced subgraphs.  ...  An intervalp-graph is the intersection graph of a collection of intervals which have been colored withpdifferent colors with edges corresponding to nonempty intersection of intervals from different color  ...  Since −1 and +1 are the same color, must be odd for all . If = 1 then from Figure 6 is an induced Figure 6 is an induced subgraph of { −1 , , +1 , 1 , . . . , } with 1 ≅ .  ... 
doi:10.1155/2014/143809 fatcat:443temg6rzaphmfe4na5qeqeiy

On Subgraphs Induced by Transversals in Vertex-Partitions of Graphs

Maria Axenovich
2006 Electronic Journal of Combinatorics  
If $H\notin {\cal F}$, then for any graph $G$ on at least $4k-1$ vertices, there is a $k$-coloring of vertices of $G$ avoiding totally multicolored induced subgraphs isomorphic to $H$.  ...  For a fixed graph $H$ on $k$ vertices, we investigate the graphs, $G$, such that for any partition of the vertices of $G$ into $k$ color classes, there is a transversal of that partition inducing $H$.  ...  any vertex-coloring of G with fixed number of colors, there is an induced monochromatic subgraph isomorphic to H?  ... 
doi:10.37236/1062 fatcat:53bjbicfknfard2nf2bbzilkze

A note on the Cornaz–Jost transformation to solve the graph coloring problem

Flavia Bonomo, Monia Giandomenico, Fabrizio Rossi
2013 Information Processing Letters  
In this note, we use a reduction by Cornaz and Jost from the graph (max-)coloring problem to the maximum (weighted) stable set problem in order to characterize new graph classes where the graph coloring  ...  problem and the more general max-coloring problem can be solved in polynomial time.  ...  If G has an odd hole as induced subgraph then T (G) has a hole of length 5 as induced subgraph, and if G has an odd antihole C k as induced subgraph then T (G) has an odd hole C k as induced subgraph.  ... 
doi:10.1016/j.ipl.2013.05.014 fatcat:xhfsquekyfhrdkeactdhkqn56a
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