List Coloring Research Articles

Upper bound for DP-chromatic number of a graph

DP-coloring as a generalization of list coloring was introduced recently by Dvořák and Postle. The D P -chromatic number of G , denoted by χ D P ( G ) , is the minimum number k such that G is D P − k -colorable. The variation of the Randić index R ′ ( G ) of a graph G is defined as the sum of the weights 1 m a x { d ( u ) , d ( v ) } of all edges u v of G , where d ( u ) is the degree of the vertex u in G . In this paper, we show that for any graph G of order n , χ D P ( G ) ≤ 2 R ′ ( G ) , and this bound is sharp for all n and 2 ≤ χ D P ( G ) ≤ n .

Flexible list colorings in graphs with special degeneracy conditions

AbstractFor a given , we say that a graph is ‐flexibly ‐choosable if the following holds: for any assignment of color lists of size on , if a preferred color from a list is requested at any set of vertices, then at least of these requests are satisfied by some ‐coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree that are ‐flexibly ‐choosable for some , which answers a question of Dvořák Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any , any graph of maximum degree that is not isomorphic to is ‐flexibly ‐choosable. Our fraction of is within a constant factor of being the best possible. We also show that graphs of treewidth 2 are ‐flexibly 3‐choosable, answering a question of Choi et al. [Flexibility of planar graphs‐sharpening the tools to get lists of size four, DAM 2022], and we give conditions for list assignments by which graphs of treewidth are ‐flexibly ‐choosable. We show furthermore that graphs of treedepth are ‐flexibly ‐choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well‐understood class of exceptions, 3‐connected nonregular graphs of maximum degree are flexibly ‐degenerate.

List Coloring Under Some Graph Operations

The list coloring of a graph G = G(V, E) is to color each vertex v ∈ V (G) from its color set L(v). If any two adjacent vertices have different colors, then G is properly colored. The aim of this paper is to study the list coloring of some graph operations

Robust Connectivity of Graphs on Surfaces

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Throughout recent decades, this problem has received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masa\v{r}\'ik, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph $G$, called the robust connectivity and written $\kappa_\rho(G)$, defined as the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used to show flexible choosability in non-regular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $\gamma$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [A conjecture on planar graphs, 1979], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

Improved lower bound for the list chromatic number of graphs with no Kt minor

AbstractHadwiger’s conjecture asserts that every graph without a $K_t$ -minor is $(t-1)$ -colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$ -minor has list chromatic number at most $ct$ . More specifically, they also conjectured that this holds for $c=\frac{3}{2}$ .Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$ -minor is at least $(2-o(1))t$ , and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$ . Our lower-bound examples are obtained via the probabilistic method.

Relaxed DP-3-Coloring of Planar Graphs Without Some Cycles

Dvořák and Postle introduced the concept of DP-coloring to overcome some difficulties in list coloring. Sittitrai and Nakprasit combined DP-coloring and defective list coloring to define a new coloring—relaxed DP-coloring. For relaxed DP-coloring, Sribunhung et al. proved that planar graphs without 4- and 7-cycles are DP-(0, 2, 2)-colorable. Li et al. proved that planar graphs without 4, 8-cycles or 4, 9-cycles are DP-(1, 1, 1)-colorable. Lu and Zhu proved that planar graphs without 4, 5-cycles, or 4, 6-cycles, or 4, 7-cycles are DP-(1, 1, 1)-colorable. In this paper, we show that planar graphs without 4, 6-cycles or 4, 8-cycles are DP-(0, 2, 2)-colorable.

DP-4-colorability of planar graphs without intersecting 5-cycles

DP-coloring of graphs as a generalization of list coloring was introduced by Dvořák and Postle (2018). In this paper, we show that every planar graph without intersecting 5-cycles is DP-4-colorable, which improves the result of Hu and Wu (2017), who proved that every planar graph without intersecting 5-cycles is 4-choosable, and the results of Kim and Ozeki (2018).

Однозначная списочная раскрашиваемость графа Кn2 + Kr

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. If a graph G is not uniquely k-list colorable, we also say that G has property M(k). The least integer k such that G has the property M(k) is called the m-number of G, denoted by m(G). In this paper, we characterize the unique list colorability of the graph G = Kn2 + Kr. In particular, we determine the number m(G) of the graph G = Kn2 + Kr.

A Grover Search-Based Algorithm for the List Coloring Problem

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ for $k\ge 5$. The list coloring problem asks the following more general question: given a list of available colors for each vertex in a graph, does it admit a proper coloring? We propose a quantum algorithm based on Grover search to quadratically speed up exhaustive search. Our algorithm loses in complexity to classical ones in specific restricted cases, but improves exhaustive search for cases where the lists and graphs considered are arbitrary in nature.

An improved upper bound for the dynamic list coloring of 1-planar graphs

<abstract><p>A graph is $ 1 $-planar if it can be drawn in the plane such that each of its edges is crossed at most once. A dynamic coloring of a graph $ G $ is a proper vertex coloring such that for each vertex of degree at least 2, its neighbors receive at least two different colors. The list dynamic chromatic number $ ch_{d}(G) $ of $ G $ is the least number $ k $ such that for any assignment of $ k $-element lists to the vertices of $ G $, there is a dynamic coloring of $ G $ where the color on each vertex is chosen from its list. In this paper, we show that if $ G $ is a 1-planar graph, then $ ch_{d}(G)\leq 10 $. This improves a result by Zhang and Li <sup>[<xref ref-type="bibr" rid="b16">16</xref>]</sup>, which says that every 1-planar graph $ G $ has $ ch_{d}(G)\leq 11 $.</p></abstract>

Cover and variable degeneracy

Let f be a nonnegative integer valued function on the vertex set of a graph. A graph is strictlyf-degenerate if each nonempty subgraph Γ has a vertex v such that degΓ(v)<f(v). In this paper, we define a new concept, strictly f-degenerate transversal, which generalizes list coloring, signed coloring, DP-coloring, L-forested-coloring, and (f1,f2,…,fs)-partition. A cover of a graph G is a graph H with vertex set V(H)=⋃v∈V(G)Xv, where Xv={(v,1),(v,2),…,(v,s)}; the edge set M=⋃uv∈E(G)Muv, where Muv is a matching between Xu and Xv. A vertex set R⊆V(H) is a transversal of H if |R∩Xv|=1 for each v∈V(G). A transversal R is a strictlyf-degenerate transversal if H[R] is strictly f-degenerate. The main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable result, signed degree-colorable result, and DP-degree-colorable result. We also give some structural results on critical graphs with respect to strictly f-degenerate transversal. Using these results, we can uniformly prove many new and known results. In the final section, we pose some open problems.

Colorings, transversals, and local sparsity

AbstractMotivated both by recently introduced forms of list coloring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi‐random method to prove the following result. For any function satisfying as , there is a function satisfying as such that the following holds. For any graph H and any partition of its vertices into parts of size at least such that (a) for each part the average over its vertices of degree to other parts is at most d, and (b) the maximum degree from a vertex to some other part is at most , there is guaranteed to be a transversal of the parts that forms an independent set of H. This is a common strengthening of two results of Loh and Sudakov (2007) and Molloy and Thron (2012), each of which in turn implies an earlier result of Reed and Sudakov (2002).

Signed colouring and list colouring of k‐chromatic graphs

AbstractA ‐colouring of a signed graph is a mapping such that for each edge , where is a symmetric integer set of size (i.e., implies that ). The signed chromatic number of a graph is the minimum integer such that for any signature of has a ‐colouring. Let be the maximum signed chromatic number of an ‐vertex ‐chromatic graph. This paper determines the value of for all positive integers . Then we study the list colouring of signed graphs. A list assignment of is called symmetric if is a symmetric integer set for each vertex . The weak signed choice number of a graph is defined to be the minimum integer such that for any symmetric ‐list assignment of , for any signature on , there is a proper ‐colouring of . We prove that the difference can be arbitrarily large. On the other hand, is bounded from above by twice the list vertex arboricity of . Using this result, we prove that , where is the complete ‐partite graph with each partite set of size 2.

Asymmetric List Sizes in Bipartite Graphs

Given a bipartite graph with parts A and B having maximum degrees at most Delta _A and Delta _B, respectively, consider a list assignment such that every vertex in A or B is given a list of colours of size k_A or k_B, respectively. We prove some general sufficient conditions in terms of Delta _A, Delta _B, k_A, k_B to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where Delta _A=Delta _B=Delta , k_A=log Delta and k_B=(1+o(1))Delta /log Delta as Delta rightarrow infty . This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteedif k_A ge Delta _A^varepsilon and k_B ge Delta _B^varepsilon for any varepsilon >0 provided Delta _A and Delta _B are large enough;if k_A ge C log Delta _B and k_B ge C log Delta _A for some absolute constant C>1; orif Delta _A=Delta _B = Delta and k_B ge C (Delta /log Delta )^{1/k_A}log Delta for some absolute constant C>0. These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.

LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

DP-4-coloring of planar graphs with some restrictions on cycles

DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring. It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7) are stated in the “color extendability” form, and uniformly proved by vertex identification and discharging method.

Planar graphs without intersecting 5-cycles are signed-4-choosable

A signed graph is a pair [Formula: see text], where [Formula: see text] is a graph and [Formula: see text] is a signature of [Formula: see text]. A set [Formula: see text] of integers is symmetric if [Formula: see text] implies that [Formula: see text]. Given a list assignment [Formula: see text] of [Formula: see text], an [Formula: see text]-coloring of a signed graph [Formula: see text] is a coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for each [Formula: see text] and [Formula: see text] for every edge [Formula: see text]. The signed choice number [Formula: see text] of a graph [Formula: see text] is defined to be the minimum integer [Formula: see text] such that for any [Formula: see text]-list assignment [Formula: see text] of [Formula: see text] and for any signature [Formula: see text] on [Formula: see text], there is a proper [Formula: see text]-coloring of [Formula: see text]. List signed coloring is a generalization of list coloring. However, the difference between signed choice number and choice number can be arbitrarily large. Hu and Wu [Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792] showed that every planar graph without intersecting 5-cycles is 4-choosable. In this paper, we prove that [Formula: see text] if [Formula: see text] is a planar graph without intersecting 5-cycles, which extends the main result of [D. Hu and J. Wu, Planar graphs without intersecting [Formula: see text]-cycles are [Formula: see text]-choosable, Discrete Math. 340 (2017) 1788–1792].

Uniquely List Colorability of Complete Split Graphs

Uniquely List Colorability of Complete Split Graphs

Generalized List Colouring of Graphs

This paper disproves a conjecture in Wang et al. (Graphs Comb. 31:1779–1787, 2015) and answers in the negative a question in Dvořák et al. (Electron J Comb:P26, 2019). In return, we pose five open problems.

Nonrepetitive List Colorings of the Integers

A coloring of the integers is nonrepetitive if no two adjacent intervals have the same color sequence. A beautiful theorem of Thue asserts that there exists a nonrepetitive coloring of {mathbb {N}} using only three colors. We obtain some generalizations of this result in which the adjacency of intervals is specified by more general graphs. We focus on the list variant of the problem, in which every integer gets a color from its own set of colors. For instance, we prove that there exists a coloring of {mathbb {N}} from arbitrary lists of size 8, such that the following property holds for every nge 1: among any 2^n consecutively adjacent intervals, each of length n, no two have the same color sequence. Another result is related to the possible extension of the famous Dejean’s conjecture to the list setting. It asserts that for every kge 1, there is a coloring of {mathbb {N}} from lists of size k+2sqrt{k}, such that no two among any k consecutively adjacent intervals have the same color sequence.