List colorings with measurable sets

The 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 such that each c(v) has measure one and 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     

Edge list multicoloring trees: An extension of Hall's theorem

We prove a necessary and sufficient condition for the existence of edge list multicoloring of trees. The result extends the Halmos–Vaughan generalization of Hall's theorem on the existence of distinct representatives of sets. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 246–255, 2003     

List coloring triangle-free planar graphs

This paper proves the following result. Assume $G$ is a triangle-free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\text{◂=▸}|L\left(v\right)|=4$ for each vertex $\text{◂+▸}v\in V\left(G\right)-X$ and $\text{◂=▸}|L\left(v\right)|=3$ for each vertex $v\in X$, then $G$ is $L$-colorable.     

Brooks' theorem with forbidden colors

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if is a connected graph with maximum degree that is not a complete graph and is a set of vertices where either

• (i) at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 4, or
• (ii) at most $$ colors are forbidden for every vertex in $$, and any two vertices of $$ are at distance at least 3,

then there is a proper $$-coloring of $$ respecting these constraints. In fact, we shall prove that these results hold in the more general setting of list colorings. These results are sharp.     

List multicoloring problems involving the k‐fold Hall numbers

We show that the four‐cycle has a k‐fold list coloring if the lists of colors available at the vertices satisfy the necessary Hall's condition, and if each list has length at least ?5k/3?; furthermore, the same is not true with shorter list lengths. In terms of h^(k)(G), the k ‐fold Hall number of a graph G, this result is stated as h^(k)(C₄)=2k??k/3?. For longer cycles it is known that h^(k)(C_n)=2k, for n odd, and 2k??k/(n?1)?≤h^(k)(C_n)≤2k, for n even. Here we show the lower bound for n even, and conjecture that this is the right value (just as for C₄). We prove that if G is the diamond (a four‐cycle with a diagonal), then h^(k)(G)=2k. Combining these results with those published earlier we obtain a characterization of graphs G with h^(k)(G)=k. As a tool in the proofs we obtain and apply an elementary generalization of the classical Hall–Rado–Halmos–Vaughan theorem on pairwise disjoint subset representatives with prescribed cardinalities. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 16–34, 2010.     

Coloring graphs from lists with bounded size of their union

A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 2^4k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory     

List edge and list total coloring of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that each 1-planar graph with maximum degree Δ is (Δ+1)-edge-choosable and (Δ+2)- total-choosable if Δ≥16, and is Δ-edge-choosable and (Δ+1)-total-choosable if Δ≥21. The second conclusion confirms the list coloring conjecture for the class of 1-planar graphs with large maximum degree.     

List Coloring with a Bounded Palette

Král' and Sgall (J Graph Theory 49(3) (2005), 177–186) introduced a refinement of list coloring where every color list must be subset to one predetermined palette of colors. We call this ‐choosability when the palette is of size at most ? and the lists must be of size at least k . They showed that, for any integer , there is an integer , satisfying as , such that, if a graph is ‐choosable, then it is C‐choosable, and asked if C is required to be exponential in k . We demonstrate it must satisfy . For an integer , if is the least integer such that a graph is ‐choosable if it is ‐choosable, then we more generally supply a lower bound on , one that is super‐polynomial in k if , by relation to an extremal set theoretic property. By the use of containers, we also give upper bounds on that improve on earlier bounds if .     

Coupled choosability of plane graphs

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying for every , there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008     

Hall's theorem for hypergraphs

We prove a hypergraph version of Hall's theorem. The proof is topological. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 83–88, 2000     

Coloring graphs from random lists of fixed size

Let G = G(n) be a graph on n vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as ) of the existence of a proper coloring ? of G, such that for every vertex v of G. We show, for all fixed k and growing n, that if , then the probability that G has such a proper coloring tends to 1 as . A similar result for complete graphs is also obtained: if and L is a random ‐list assignment for the complete graph K_n on n vertices, then the probability that K_n has a proper coloring with colors from the random lists tends to 1 as .Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317‐327, 2014     

Equitable List Coloring of Graphs with Bounded Degree

A graph G is equitably k‐choosable if for every k‐list assignment L there exists an L‐coloring of G such that every color class has at most vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably ‐choosable. In particular, we confirm the conjecture for and show that every graph with maximum degree at most r and at least r³ vertices is equitably ‐choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings.     

最大度为4的图的无圈列表边染色

对于图G=(V(G),E(G)),如果一个映射φ:E(G)→{1,2,…,k},使得G中任意相邻的两边e₁,e₂满足φ(e₁)≠φ(e₂),并且G中不含有双色圈,则称φ为G的一个无圈边染色.对于给定的列表分配L={L(e)|e∈E(G)},如果存在图G的一个无圈边染色φ,使得对于任意边e∈E(G),均有φ(e)∈L(e),则称染色φ为G的一个无圈L-边染色.如果对于任意的列表分配L,当对所有的边e∈E(G)满足|L(e)|≥k时,图G均存在无圈L-边染色,那么称G是无圈k-边可选的.使图G无圈k-边可选的最小的正整数k,称为G的无圈列表边色数,用a’_l(G)表示.本文证明了对于最大度△≤4的连通图G,如果|E(G)|≤2|V(G)|-1,则a’_l(G)≤6,扩展了Basavaraju和Chandran文[J.Graph Theory,2009,61(3):192-209]的结果.     

Hall parameters of complete and complete bipartite graphs

Given a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well‐known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ? with ?(υ) ∈ L (υ) (?υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non‐triviality condition, called Hall's condition, has to be satisfied by the list assignment. The edge‐analogue of the Hall number is called the Hall index, h′(G), and the total analogue is called the total Hall number, h″(G), of G. If the stock of colours from which L(υ) is selected is restricted to a set of size k, then the analogous numbers are called k‐restricted, or restricted, Hall parameters, and are denoted by h_k(G), h′_k(G) and h″_k(G). Our main object in this article is to determine, or closely bound, h′(K), h″(K_n), h′(K_m,n) and h′_k(K_m,n). We also answer some hitherto unresolved questions about Hall parameters. We show in particular that there are examples of graphs G with h′(G)?h′(G ? e)>1. We show that there are examples of graphs G and induced subgraphs H with h_k(G)<h_k(H) [this phenomenon cannot occur with unrestricted Hall numbers]. We also give an example of a graph G and an integer k such that h_k(G)<χ(G)<h(G). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 208–237, 2002     

Coloring graphs of various maximum degree from random lists

Let G = G(n) be a graph on n vertices with maximum degree Δ =Δ (n). Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all k‐subsets of a color set of size . Such a list assignment is called a random ‐list assignment. In this paper, we are interested in determining the asymptotic probability (as n →∞) of the existence of a proper coloring φ of G, such that for every vertex v of G, a so‐called L‐coloring. We give various lower bounds on σ, in terms of n, k, and Δ, which ensures that with probability tending to 1 as n →∞ there is an L‐coloring of G. In particular, we show, for all fixed k and growing n, that if and , then the probability that G has an L‐coloring tends to 1 as . If and , then the same conclusion holds provided that . We also give related results for other bounds on Δ, when k is constant or a strictly increasing function of n.     

List coloring with requests

Let be a graph with a list assignment . Suppose a preferred color is given for some of the vertices; how many of these preferences can be respected when -coloring ? We explore several natural questions arising in this context, and propose directions for further research.     

Theorem of Kuratowski-Suslin for measurable mappings. II

The purpose of this paper is to describe these -measurable mappings on a separable complete metric space with the Borel measure , which transform every -measurable set onto a -measurable one. The obtained results are a generalization of the classical outcomes of Suslin and Kuratowski and the results from our previous paper.

     

Dynamic choosability of triangle‐free graphs and sparse random graphs

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).