List Coloring Research Articles

Coloring uniform hypergraphs with few edges

AbstractA hypergraph is b‐simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r‐uniform non‐t‐colorable hypergraph of girth at least g.Erdős and Lovász proved that A result of Szabó improves the lower bound by a factor of r2−ϵ for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b‐simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and ϵ > 0 and sufficiently large r, every r‐uniform b‐simple hypergraph $\cal {H}$ with maximum edge‐degree at most trr1−ϵ is t‐colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Mr. Paint and Mrs. Correct

We introduce a coloring game on graphs, in which each vertex $v$ of a graph $G$ owns a stack of $\ell_v{-}1$ erasers. In each round of this game the first player Mr. Paint takes an unused color, and colors some of the uncolored vertices. He might color adjacent vertices with this color – something which is considered "incorrect". However, Mrs. Correct is positioned next to him, and corrects his incorrect coloring, i.e., she uses up some of the erasers – while stocks (stacks) last – to partially undo his assignment of the new color. If she has a winning strategy, i.e., she is able to enforce a correct and complete final graph coloring, then we say that $G$ is $\ell$-paintable. Our game provides an adequate game-theoretic approach to list coloring problems. The new concept is actually more general than the common setting with lists of available colors. It could have applications in time scheduling, when the available time slots are not known in advance. We give an example that shows that the two notions are not equivalent; $\ell$-paintability is stronger than $\ell$-list colorability. Nevertheless, many deep theorems about list colorability remain true in the context of paintability. We demonstrate this fact by proving strengthened versions of classical list coloring theorems. Among the obtained extensions are paintability versions of Thomassen's, Galvin's and Shannon's Theorems.

Hall number for list colorings of graphs: Extremal results

Given a graph G = ( V , E ) and sets L ( v ) of allowed colors for each v ∈ V , a list coloring of G is an assignment of colors φ ( v ) to the vertices, such that φ ( v ) ∈ L ( v ) for all v ∈ V and φ ( u ) ≠ φ ( v ) for all u v ∈ E . The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L ( v ) | ≥ k holds for every vertex v . This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L ( v ) . (On complete graphs, this condition is equivalent to the well-known one in Hall’s Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n − 3 . This estimate is tight for all n . Tightness is deduced from the upper bound that every graph of order n has Hall number at most n − 2 . We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K 2 ∪ ( n − 2 ) K 1 , P 4 ∪ ( n − 4 ) K 1 , and C 5 ∪ ( n − 5 ) K 1 . Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161–182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227–245] as a function of maximum degree.

Equitable list coloring and treewidth

AbstractGiven lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most ⌈|V(G)|/k⌉ vertices. A graph is equitably k ‐choosable if such a coloring exists whenever the lists all have size k. Kostochka, Pelsmajer, and West introduced this notion and conjectured that G is equitably k‐choosable for k>Δ(G). We prove this for graphs of treewidth w≤5 if also k≥3w−1. We also show that if G has treewidth w≥5, then G is equitably k‐choosable for k≥max{Δ(G)+w−4, 3w−1}. As a corollary, if G is chordal, then G is equitably k‐choosable for k≥3Δ(G)−4 when Δ(G)>2. © 2009 Wiley Periodicals, Inc. J Graph Theory

Adapted list coloring of planar graphs

AbstractGiven an edge coloring F of a graph G, a vertex coloring of G is adapted to F if no color appears at the same time on an edge and on its two endpoints. If for some integer k, a graph G is such that given any list assignment L to the vertices of G, with |L(v)|⩾k for all v, and any edge coloring F of G, G admits a coloring c adapted to F where c(v)∈L(v) for all v, then G is said to be adaptably k‐choosable. In this note, we prove that K5‐minor‐free graphs are adaptably 4‐choosable, which implies that planar graphs are adaptably 4‐colorable and answers a question of Hell and Zhu. We also prove that triangle‐free planar graphs are adaptably 3‐choosable and give negative results on planar graphs without 4‐cycle, planar graphs without 5‐cycle, and planar graphs without triangles at distance t, for any t⩾0. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 127–138, 2009

Precoloring extension for K4‐minor‐free graphs

AbstractLet G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W⊆V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K4‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009

Planar Graphs without 7-Cycles Are 4-Choosable

Previous article Next article Planar Graphs without 7-Cycles Are 4-ChoosableBabak FarzadBabak Farzadhttps://doi.org/10.1137/05064477XPDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper, we show that every planar graph without 7-cycles is 4-choosable.[1] S. Choudum, Some 4-valent, 3-connected, planar, almost pancyclic graphs, Discrete Math., 18 (1977), pp. 125–129. DSMHA4 0012-365X CrossrefISIGoogle Scholar[2] G. Fijavz, , M. Juvan, , B. Mohar and , and R. Skrekovski, Planar graphs without cycles of specific lengths, European J. Combin., 23 (2002), pp. 377–388. EJOCDI 0195-6698 CrossrefISIGoogle Scholar[3] S. Gutner, The complexity of planar graph choosability, Discrete Math., 159 (1996), pp. 119–130. DSMHA4 0012-365X CrossrefISIGoogle Scholar[4] P. Lam, , W. Shiu and , and B. Xu, On structure of some plane graphs with application to choosability, J. Combin. Theory Ser. B, 82 (2001), pp. 285–296. 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AMLEEL 0893-9659 CrossrefISIGoogle ScholarKeywordsplanar graphscoloringchoosability Previous article Next article FiguresRelatedReferencesCited byDetails A sufficient condition for planar graphs to be DP -4-colorable7 October 2022 | Quaestiones Mathematicae, Vol. 5 Cross Ref Two sufficient conditions for a planar graph to be list vertex-2-arborableDiscrete Mathematics, Vol. 345, No. 6 Cross Ref Cover and variable degeneracyDiscrete Mathematics, Vol. 345, No. 4 Cross Ref DP-4-coloring of planar graphs with some restrictions on cyclesDiscrete Mathematics, Vol. 344, No. 11 Cross Ref DP-coloring on planar graphs without given adjacent short cycles10 October 2020 | Discrete Mathematics, Algorithms and Applications, Vol. 13, No. 02 Cross Ref Planar Graphs Without Pairwise Adjacent $3$-, $4$-, $5$-, and $6$-cycle are $4$-choosableTaiwanese Journal of Mathematics, Vol. -1, No. -1 Cross Ref Planar graphs without 7-cycles and butterflies are DP-4-colorableDiscrete Mathematics, Vol. 343, No. 8 Cross Ref DP-4-colorability of planar graphs without adjacent cycles of given lengthDiscrete Applied Mathematics, Vol. 277 Cross Ref List coloring and diagonal coloring for plane graphs of diameter twoApplied Mathematics and Computation, Vol. 363 Cross Ref Facial Colorings of Plane Graphs28 April 2019 | Journal of Interconnection Networks, Vol. 19, No. 01 Cross Ref Planar graphs without chordal 6-cycles are 4-choosableDiscrete Applied Mathematics, Vol. 244 Cross Ref Planar graphs without intersecting 5 -cycles are 4 -choosableDiscrete Mathematics, Vol. 340, No. 8 Cross Ref (4, 2)-Choosability of Planar Graphs with Forbidden Structures14 June 2017 | Graphs and Combinatorics, Vol. 33, No. 4 Cross Ref A sufficient condition for a planar graph to be 4 -choosableDiscrete Applied Mathematics, Vol. 224 Cross Ref Toroidal graphs containing neither K5− nor 6-cycles are 4-choosable26 May 2016 | Journal of Graph Theory, Vol. 85, No. 1 Cross Ref Planar graphs without 4-cycles adjacent to triangles are 4-choosableDiscrete Mathematics, Vol. 339, No. 12 Cross Ref A Note on 3-choosability of Planar Graphs Related to Montanssier’s Conjecture20 November 2018 | Canadian Mathematical Bulletin, Vol. 59, No. 2 Cross Ref Improper Choosability of Planar Graphs without 4-CyclesYingqian Wang and Lingji Xu5 December 2013 | SIAM Journal on Discrete Mathematics, Vol. 27, No. 4AbstractPDF (192 KB)On Coloring Problems26 July 2013 Cross Ref On the vertex-arboricity of planar graphs without 7-cyclesDiscrete Mathematics, Vol. 312, No. 15 Cross Ref Choosability of toroidal graphs without short cyclesJournal of Graph Theory, Vol. 28 Cross Ref Volume 23, Issue 3| 2009SIAM Journal on Discrete Mathematics History Submitted:29 November 2005Accepted:06 April 2009Published online:15 July 2009 InformationCopyright © 2009 Society for Industrial and Applied MathematicsKeywordsplanar graphscoloringchoosabilityMSC codes05C1505C10PDF Download Article & Publication DataArticle DOI:10.1137/05064477XArticle page range:pp. 1179-1199ISSN (print):0895-4801ISSN (online):1095-7146Publisher:Society for Industrial and Applied Mathematics

Set Partitioning via Inclusion-Exclusion

Given a set $N$ with $n$ elements and a family $\mathcal{F}$ of subsets, we show how to partition $N$ into $k$ such subsets in $2^n n^{O(1)}$ time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in $2^n n^{O(1)}$ time for several well-studied partition problems including domatic number, chromatic number, maximum $k$-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is available, our algorithms run in time $3^n n^{O(1)}$ if membership in $\mathcal{F}$ can be decided in polynomial time. We solve chromatic number in $O(2.2461^n)$ time and domatic number in $O(2.8718^n)$ time. Finally, we present a family of polynomial space approximation algorithms that find a number between $\chi(G)$ and $\lceil(1+\epsilon)\chi(G)\rceil$ in time $O(1.2209^n+2.2461^{e^{-\epsilon}n})$.

The chromatic polynomial and list colorings

We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ l ( G ) denote the list chromatic number of G . In this paper we prove that for every graph G for which the length of each cycle is divisible by l ( l ≥ 3 ), χ l ( G ) ≤ 3 .

List colorings with measurable sets

AbstractThe measurable list chromatic number of a graph G is the smallest number ξ such that if each vertex v of G is assigned a set L(v) of measure ξ in a fixed atomless measure space, then there exist sets $c(v)\subseteq L(v)$ such that each c(v) has measure one and $c(v)\cap c(v') = \emptyset$ for every pair of adjacent vertices v and v'. We provide a simpler proof of a measurable generalization of Hall's theorem due to Hilton and Johnson [J Graph Theory 54 (2007), 179–193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 229–238, 2008

Precoloring extension for 2-connected graphs with maximum degree three

Let G = G ( V , E ) be a simple graph, L a list assignment with | L ( v ) | = Δ ( G ) ∀ v ∈ V and W ⊆ V an independent subset of the vertex set. Define d ( W ) ≔ min { d ( v , w ) ∣ v , w ∈ W } to be the minimum distance between two vertices of W . In this paper it is shown that if G is 2-connected with Δ ( G ) = 3 and G is not K 4 then every precoloring of W is extendable to a proper list coloring of G provided that d ( W ) ≥ 6 . An example shows that the bound is sharp. This result completes the investigation of precoloring extensions for graphs with | L ( v ) | = Δ ( G ) for all v ∈ V where the precolored set W is an independent set.

Equitable list colorings of planar graphs without short cycles

A graph G is equitably k -choosable if, for any k -uniform list assignment L , G is L -colorable and each color appears on at most ⌈ ∣ V ( G ) ∣ k ⌉ vertices. A graph G is equitably k -colorable if G has a proper k -vertex coloring such that the sizes of any two color classes differ by at most 1. In this paper, we prove that every planar graph G is equitably k -choosable and equitably k -colorable if one of the following conditions holds: (1) G is triangle-free and k ≥ max { Δ ( G ) , 8 } ; (2) G has no 4- and 5-cycles and k ≥ max { Δ ( G ) , 7 } .

Complexity of unique list colorability

Given a list L ( v ) for each vertex v , we say that the graph G is L - colorable if there is a proper vertex coloring of G where each vertex v takes its color from L ( v ) . The graph is uniquely k - list colorable if there is a list assignment L such that ∣ L ( v ) ∣ = k for every vertex v and the graph has exactly one L -coloring with these lists. Mahdian and Mahmoodian [M. Mahdian, E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295–305] gave a polynomial-time characterization of uniquely 2-list colorable graphs. Answering an open question from [M. Ghebleh, E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. 59 (2001) 307–318; M. Mahdian, E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295–305], we show that uniquely 3-list colorable graphs are unlikely to have such a nice characterization, since recognizing these graphs is Σ 2 p -complete.

Equitable list coloring of planar graphs without 4- and 6-cycles

A graph G is equitably k -choosable if for any k -uniform list assignment L , there exists an L -colorable of G such that each color appears on at most ⌈ | V ( G ) | k ⌉ vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k -choosable for k > Δ ( G ) . We prove this for planar graphs with Δ ( G ) ≥ 6 and no 4- or 6-cycles.

List Colouring Constants of Triangle Free Graphs

Abstract In this paper we prove a result about vertex list colourings which in particular shows that a conjecture of the second author (1999, Journal of Graph Theory 31, 149-153) is true for triangle free graphs of large maximum degree. There exists a constant K such that the following holds: Given a graph G and a list assignment L to vertices of G, assigning a list of available colours L ( v ) to each vertex v ∈ V ( G ) , such that | L ( v ) | = K Δ log ( Δ ) , then there exists a proper list colouring of vertices of G provided that for each colour c, the graph induced by all vertices v with c ∈ L ( v ) is triangle free and has maximum degree at most Δ.

On the Complexity of Ordered Colorings

We introduce two variants of proper colorings with imposed partial ordering on the set of colors. One variant shows very close connections to some fundamental problems in graph theory, e.g., directed graph homomorphism and list colorings. We study the border between tractability and intractability for both variants.

Adapted List Coloring of Graphs and Hypergraphs

We introduce and study adapted list coloring of graphs and hypergraphs. This is a generalization of ordinary list coloring and adapted coloring, and has more applications than these. We prove that the upper bounds on the adaptable choosability of graphs and uniform hypergraphs in terms of maximum degree are sufficiently stronger than those on the ordinary choosability, while the bounds in terms of degeneracy are the same. We also characterize simple graphs with adaptable choosability $2$.

The k-fold list coloring of cycles with Hall's condition

We prove that any cycle Cn, n ≥ 4, with list assignment L, has a k -fold list coloring from the given lists if (i) each list contains at least 2k colors and (ii) Cn and L satisfy Hall's condition for k -fold list colorings. Further, 2k in (i) cannot be replaced by 2k - 1 if either n is odd, or n is even and n ≥ k + 2. In other words, if n ≥ 4, the k -fold Hall number of a cycle Cn satifies h(k)(Cn) ≤ 2k, with equality if n is odd, or n is even and n ≥ k + 2.

Brooks-Type Theorems for Pair-List Colorings and List Homomorphisms

Brooks proved that every connected graph other than a clique or odd cycle can be colored with $\Delta$ colors. Erdos, Rubin, and Taylor (and, independently, Vizing) generalized the theorem of Brooks to list colorings, describing all uncolorable connected graphs in which no vertex has a list smaller than its degree. Other authors have extended this to list $T$-colorings and their generalizations. We further extend it to model pair-list colorings. In addition to including all of the previous situations, pair-list colorings also generalize list homomorphisms (also known as list $H$-colorings). In the general context of pair-list colorings, we prove a Brooks-type theorem which extends many (but not all) of the existing results. Our result applies to both graphs and digraphs, with or without loops. We discuss several applications of the result, including a polynomial test for the existence of balanced list homomorphisms and retractions.