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

2004
*
Electronic Journal of Combinatorics
*

We prove several other structural

doi:10.37236/1779
fatcat:tdjdfhtp65fuvor4mzuywr6x6e
*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. ...##
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Algorithms
[chapter]

2011
*
Graph Coloring Problems
*

3-Chromatic Uniform Cliques Monochromatic Sum-Sets Arithmetic Progressions Unprovability The

doi:10.1002/9781118032497.ch10
fatcat:374tktuvgvekni4fnz3dgbytjm
*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 ...##
###
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. ...##
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Parameterized algorithms for feedback set problems and their duals in tournaments

2006
*
Theoretical Computer Science
*

More specifically, the dual problem we show fixed parameter tractable are: Given an oriented

doi:10.1016/j.tcs.2005.10.010
fatcat:2k3itfzss5corhrwpdwamcjiry
*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. ...##
###
Induced subgraphs of induced subgraphs of large chromatic number
[article]

2022
*
arXiv
*
pre-print

We prove that, for every

arXiv:2203.03612v2
fatcat:n4a7wstixzhwnjrmprtowhsuam
*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 ...##
###
Acyclic graphs with at least 2ℓ+1 vertices are ℓ-recognizable
[article]

2021
*
arXiv
*
pre-print

The (n-ℓ)-deck

arXiv:2103.12153v1
fatcat:3t7jdm5d7nf6thl3c3d5dukefm
*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 . ...##
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An Introduction to the Discharging Method via Graph Coloring
[article]

2016
*
arXiv
*
pre-print

Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique

arXiv:1306.4434v2
fatcat:znrp36yihfcwfj2fcxy3hdaf6y
*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. ...##
###
Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth
[chapter]

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 ∞. ...

##
###
An introduction to the discharging method via graph coloring

2017
*
Discrete Mathematics
*

Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique

doi:10.1016/j.disc.2016.11.022
fatcat:neapvvzudbd2th7wv5qj5kf4ea
*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. ...##
###
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]

2015
*
arXiv
*
pre-print

Recently, Borodin, Kostochka,

arXiv:1507.02815v1
fatcat:7a75jrmcn5g3bkijf7lizqd5wq
*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 ...##
###
Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

2016
*
Journal of Graph Theory
*

Recently, Borodin, Kostochka,

doi:10.1002/jgt.22093
fatcat:b2i4viue75dojn2abbyayo7rtm
*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 ...##
###
From the plane to higher surfaces

2012
*
Journal of combinatorial theory. Series B (Print)
*

These include the 5-

doi:10.1016/j.jctb.2012.03.001
fatcat:kvckd6thxre7nnvtt3tf5vdtai
*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. ...##
###
The m-Degenerate Chromatic Number of a Digraph
[article]

2018
*
arXiv
*
pre-print

an

arXiv:1409.7535v2
fatcat:znz7ttdz2vh7pmbflrxyjh5abe
*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. ...##
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Oriented colorings of triangle-free planar graphs

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 ...

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