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Coloring with no $2$-Colored $P_4$'s

Michael O. Albertson, Glenn G. Chappell, H. A. Kierstead, André Kündgen, Radhika Ramamurthi
2004 Electronic Journal of Combinatorics  
We prove several other structural and topological results for star colorings, such as: cubic graphs are $7$-colorable, and planar graphs of girth at least $7$ are $9$-colorable.  ...  Star colorings are a strengthening of acyclic colorings, i.e., proper colorings in which every two color classes induce a forest.  ...  Using other results from acyclic coloring, we also improve bounds on the star chromatic number for planar graphs of girth at least 5 and 7 mentioned in [6] . Proof.  ... 
doi:10.37236/1779 fatcat:tdjdfhtp65fuvor4mzuywr6x6e

Algorithms [chapter]

2011 Graph Coloring Problems  
3-Chromatic Uniform Cliques Monochromatic Sum-Sets Arithmetic Progressions Unprovability The Direct Product of Hypergraphs Maximal Complete Subgraphs in Perfect Graphs Coloring Triangulable Manifolds  ...  Graphs 122 7.2 Grunbaum's Girth Problem 123 7.3 Smallest Triangle-Free/t-Chromatic Graphs 123 7.4 Large Bipartite Subgraphs of Triangle-Free Graphs 126 7.5 Sparse Subgraphs 126 7.6 Number  ... 
doi:10.1002/9781118032497.ch10 fatcat:374tktuvgvekni4fnz3dgbytjm

Page 1323 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
The main result of this paper is that every planar graph of girth at least 5 is 3-choosable and that one may precolor any 5-cycle in the graph.  ...  Planar directed ordered connected acyclic graphs (pdags) built over a finite doubly ranked alphabet by parallel and serial compo- sition are studied.  ... 

Parameterized algorithms for feedback set problems and their duals in tournaments

Venkatesh Raman, Saket Saurabh
2006 Theoretical Computer Science  
More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of k vertices (arcs) that forms an acyclic directed subgraph of the graph?  ...  /k) k k 2 +min{m lg n, n 2 }) algorithm for the dual of feedback vertex set problem (maximum vertex induced acyclic graph) in oriented directed graphs, and an O(4 k k + m) algorithm for the dual of feedback  ...  Srinivasa Rao for fruitful discussions and comments on earlier versions of this paper and the two anonymous referees for several useful comments.  ... 
doi:10.1016/j.tcs.2005.10.010 fatcat:2k3itfzss5corhrwpdwamcjiry

Induced subgraphs of induced subgraphs of large chromatic number [article]

António Girão, Freddie Illingworth, Emil Powierski, Michael Savery, Alex Scott, Youri Tamitegama, Jane Tan
2022 arXiv   pre-print
We prove that, for every graph F with at least one edge, there is a constant c_F such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced  ...  This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Moreover, we show an analogous statement where clique number is replaced by odd girth.  ...  As the properties listed above are the only properties that we will require, one could in fact use any sequence of directed graphs ( # » X n ) and corresponding sequence (X n ) of underlying undirected  ... 
arXiv:2203.03612v2 fatcat:n4a7wstixzhwnjrmprtowhsuam

Acyclic graphs with at least 2ℓ+1 vertices are ℓ-recognizable [article]

Alexandr V. Kostochka, Mina Nahvi, Douglas B. West, Dara Zirlin
2021 arXiv   pre-print
The (n-ℓ)-deck of an n-vertex graph is the multiset of subgraphs obtained from it by deleting ℓ vertices.  ...  A family of n-vertex graphs is ℓ-recognizable if every graph having the same (n-ℓ)-deck as a graph in the family is also in the family.  ...  When F is a class of graphs, an F -subgraph of a graph G is an induced subgraph of G in F . Let s(F, G) denote the number of induced subgraphs of G isomorphic to F .  ... 
arXiv:2103.12153v1 fatcat:3t7jdm5d7nf6thl3c3d5dukefm

An Introduction to the Discharging Method via Graph Coloring [article]

Daniel W. Cranston, Douglas B. West
2016 arXiv   pre-print
Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples.  ...  We provide a "how-to" guide to the use and application of the Discharging Method.  ...  An acyclic coloring of a graph is a proper coloring such that the union of any two color classes induces an acyclic subgraph; equivalently, no cycle is 2-colored. Theorem 2.3.  ... 
arXiv:1306.4434v2 fatcat:znrp36yihfcwfj2fcxy3hdaf6y

Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth [chapter]

Kazuhiro Kurita, Kunihiro Wasa, Alessio Conte, Takeaki Uno, Hiroki Arimura
2018 Lecture Notes in Computer Science  
In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph.  ...  We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in O(n) amortized time per solution and using O(n^3) space.  ...  For simplicity, we say that G has girth k if g(G) ≥ k. The girth of acyclic graphs is usually assumed to be ∞.  ... 
doi:10.1007/978-3-319-94667-2_17 fatcat:rc6knbovwrb6zb3kejc4g5pyqe

An introduction to the discharging method via graph coloring

Daniel W. Cranston, Douglas B. West
2017 Discrete Mathematics  
Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples.  ...  We provide a "how-to" guide to the use and application of the Discharging Method.  ...  An acyclic coloring of a graph is a proper coloring such that the union of any two color classes induces an acyclic subgraph; equivalently, no cycle is 2-colored. Theorem 2.3.  ... 
doi:10.1016/j.disc.2016.11.022 fatcat:neapvvzudbd2th7wv5qj5kf4ea

Page 576 of Mathematical Reviews Vol. 46, Issue 3 [page]

1973 Mathematical Reviews  
Author’s introduction: ‘We consider the problem of enumerating those spanning trees of a graph that contain a given acyclic subgraph. 8S. L.  ...  Let g(d, t, v, n) be a connected graph of order n, diameter d, girth 2d, in which each pair of nodes at distance d is connected by ¢ distinct paths of length d (which is proved 3367 576 COMBINATORICS to  ... 

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths [article]

Maria Axenovich and Torsten Ueckerdt and Pascal Weiner
2015 arXiv   pre-print
Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs.  ...  Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching.  ...  Conclusions and open questions In this paper, we proved that for any planar graph of girth 6 and any assignment of lists of 2 colors to each vertex, there is a coloring from these lists such that monochromatic  ... 
arXiv:1507.02815v1 fatcat:7a75jrmcn5g3bkijf7lizqd5wq

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

Maria Axenovich, Torsten Ueckerdt, Pascal Weiner
2016 Journal of Graph Theory  
Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs.  ...  Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching.  ...  Conclusions and open questions In this paper, we proved that for any planar graph of girth 6 and any assignment of lists of 2 colors to each vertex, there is a coloring from these lists such that monochromatic  ... 
doi:10.1002/jgt.22093 fatcat:b2i4viue75dojn2abbyayo7rtm

From the plane to higher surfaces

Ken-ichi Kawarabayashi, Carsten Thomassen
2012 Journal of combinatorial theory. Series B (Print)  
These include the 5-list-color theorem, results on arboricity, and various types of colorings, and decomposition theorems of planar graphs into two graphs with prescribed degeneracy properties.  ...  (g) vertices where f (g) is the smallest edge-width which guarantees a graph of Euler genus g and girth 5 to be 3-colorable.  ...  interest in that it may reduce extendability problems on graphs of girth 4 to graphs of girth 5.  ... 
doi:10.1016/j.jctb.2012.03.001 fatcat:kvckd6thxre7nnvtt3tf5vdtai

The m-Degenerate Chromatic Number of a Digraph [article]

Noah Golowich
2018 arXiv   pre-print
an acyclic subgraph.  ...  For any m ≥ 1, a digraph is weakly m-degenerate if each of its induced subgraphs has a vertex of in-degree or out-degree less than m.  ...  An acyclic set in a digraph is a set of vertices whose induced subgraph contains no directed cycle.  ... 
arXiv:1409.7535v2 fatcat:znz7ttdz2vh7pmbflrxyjh5abe

Oriented colorings of triangle-free planar graphs

Pascal Ochem
2004 Information Processing Letters  
In this note, we focus on planar graphs and we use the notation P k for the class of planar graphs of girth at least k.  ...  For every remaining sequence and for every orientation vector, 5 α-successors are listed in the appendix. ✷  ...  Appendix The quadratic residues of 59: 0 1 3 4 5 7 9 12 15 16 17 19 20 21 22 25 26 27 28 29 35 36 41 45 46 48 49 51 53 57 The  ... 
doi:10.1016/j.ipl.2004.06.012 fatcat:2ktjlcwwnnbhvjg3ymqek635mq
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