List Coloring Research Articles

Breaking small automorphisms by list colourings

AbstractFor a graph , we define a small automorphism as one that maps some vertex into its neighbour. We investigate the edge colourings of that break every small automorphism of . We show that such a colouring can be chosen from any set of lists of length 3. In addition, we show that any set of lists of length 2 on both edges and vertices of yields a total colouring which breaks all the small automorphisms of . These results are sharp, and they match the known bounds for the nonlist variant.

Parameterized problems complete for nondeterministic FPT time and logarithmic space

Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f(k)nO(1) and space f(k)log⁡(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloring and Precoloring Extension with pathwidth as parameter, Scheduling of Jobs with Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidth and variants of Weighted CNF-Satisfiability. In particular, this implies that all these problems are W[t]-hard for all t.

Flexible list colorings: Maximizing the number of requests satisfied

AbstractFlexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose , is a graph, is a list assignment for , and is a function with nonempty domain such that for each ( is called a request of ). The triple is ‐satisfiable if there exists a proper ‐coloring of such that for at least vertices in . We say is ‐flexible if is ‐satisfiable whenever is a ‐assignment for and is a request of . It was shown by Dvořák et al. that if is prime, is a ‐degenerate graph, and is a request for with domain of size 1, then is 1‐satisfiable whenever is a ‐assignment. In this paper, we extend this result to all for bipartite ‐degenerate graphs.The literature on flexible list coloring tends to focus on showing that for a fixed graph and there exists an such that is ‐flexible, but it is natural to try to find the largest possible for which is ‐flexible. In this vein, we improve a result of Dvořák et al., by showing ‐degenerate graphs are ‐flexible. In pursuit of the largest for which a graph is ‐flexible, we observe that a graph is not ‐flexible for any if and only if , where is the Hall ratio of , and we initiate the study of the list flexibility number of a graph , which is the smallest such that is ‐flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

A Strengthening and an Efficient Implementation of Alon-Tarsi List Coloring Method

As one of the first applications of the polynomial method in combinatorics, Alon and Tarsi proved that if certain coefficients of the graph polynomial are non-zero, then the graph is choosable, i.e., colorable from any assignment of lists of prescribed size. We show that in case all relevant coefficients are zero, then further coefficients of the graph polynomial provide constraints on the list assignments from which the graph cannot be colored. This often enables us to confirm colorability from a given list assignment, or to decide choosability by testing just a few list assignments. We also describe an efficient way to implement this approach, making it feasible to test choosability of graphs with around 70 edges.

Edge-colouring graphs with local list sizes

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ≥ln25⁡Δ, the following holds: for every assignment L of lists of colours to the edges of G, such that |L(e)|≥(1+o(1))⋅max⁡{deg⁡(u),deg⁡(v)} for each edge e=uv, there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

Kempe changes in degenerate graphs

We consider Kempe changes on the k-colorings of a graph on n vertices. If the graph is (k−1)-degenerate, then all its k-colorings are equivalent up to Kempe changes. However, the sequence between two k-colorings that arises from the proof may have length exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most k−1. Namely, any two k-colorings are equivalent up to O(kn2) Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As one of the main results, we derive that given an n-vertex graph with maximum degree Δ, the Δ-colorings are all equivalent up to OΔ(n2) Kempe changes, unless Δ=3 and some connected component is a 3-prism, that is K2□K3, in which case there exist some non-equivalent 3-colorings.

An algebraic approach for counting DP-3-colorings of sparse graphs

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial of a graph G, P(G,m), and the list color function, Pℓ(G,m), the DP-color function of G, denoted by PDP(G,m), counts the minimum number of DP-colorings over all possible m-fold covers. It follows that PDP(G,m)≤Pℓ(G,m)≤P(G,m). A function f is chromatic-adherent if for every graph G, f(G,a)=P(G,a) for some a≥χ(G) implies that f(G,m)=P(G,m) for all m≥a. It is known that the DP-color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose G is an n-vertex graph and H is a 3-fold cover of G. In this paper we associate with H a polynomial fG,H∈F3[x1,…,xn] so that the number of non-zeros of fG,H equals the number of H-colorings of G. We then use a well-known result of Alon and Füredi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on PDP(G,3) when 2n>|E(G)|. An easy consequence of this is that PDP(G,3)≥3n/6 for every n-vertex planar graph G of girth at least 5, improving the previously known bounds on both PDP(G,3) and Pℓ(G,3). Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP-color function.

Brooks' theorem with forbidden colors

AbstractWe 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 at most colors are forbidden for every vertex in , and any two vertices of are at distance at least 4, or 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.

Optimally reconfiguring list and correspondence colourings

The reconfiguration graph Ck(G) for the k-colourings of a graph G has a vertex for each proper k-colouring of G, and two vertices of Ck(G) are adjacent precisely when those k-colourings differ on a single vertex of G. Much work has focused on bounding the maximum value of diamCk(G) over all n-vertex graphs G. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if L is a list-assignment for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamCL(G)≤n(G)+μ(G). We also conjecture that if (L,H) is a correspondence cover for a graph G with |L(v)|≥d(v)+2 for all v∈V(G), then diamC(L,H)(G)≤n(G)+τ(G). (Here μ(G) and τ(G) denote the matching number and vertex cover number of G.) For every graph G, we give constructions showing that both conjectures are best possible, which also hints towards an exact form of Cereceda’s Conjecture for regular graphs. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) diamCL(G)≤n(G)+2μ(G) and diamC(L,H)(G)≤n(G)+2τ(G). Our second main result proves that both conjectured bounds hold, whenever all v satisfy |L(v)|≥2d(v)+1. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree. The full paper can also be found at arxiv.org/abs/2204.07928.

An Improved Bound for the Linear Arboricity Conjecture

In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph G of maximum degree Delta can be decomposed into at most linear forests. (A forest is linear if all of its components are paths.) In 1988, Alon proved the conjecture holds asymptotically. The current best bound is due to Ferber, Fox and Jain from 2020 who showed that frac{Delta }{2}+ O(Delta ^{0.661}) suffices for large enough Delta . Here, we show that G admits a decomposition into at most frac{Delta }{2}+ 3sqrt{Delta } log ^4 Delta linear forests provided Delta is large enough. Moreover, our result also holds in the more general list setting, where edges have (possibly different) sets of permissible linear forests. Thus our bound also holds for the List Linear Arboricity Conjecture which was only recently shown to hold asymptotically by Kim and the second author. Indeed, our proof method ties together the Linear Arboricity Conjecture and the well-known List Colouring Conjecture; consequently, our error term for the Linear Arboricity Conjecture matches the best known error-term for the List Colouring Conjecture due to Molloy and Reed from 2000. This follows as we make two copies of every colour and then seek a proper edge colouring where we avoid bicoloured cycles between a colour and its copy; we achieve this via a clever modification of the nibble method.

Sudoku number of graphs

We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let be a graph of order n with chromatic number and let Let be a k-coloring of the induced subgraph The coloring is called an extendable coloring if can be extended to a k-coloring of G. We say that is a Sudoku coloring of G if can be uniquely extended to a k-coloring of G. The smallest order of such an induced subgraph of G which admits a Sudoku coloring is called the Sudoku number of G and is denoted by In this paper we initiate a study of this parameter. We first show that this parameter is related to list coloring of graphs. In Section 2, basic properties of Sudoku coloring that are related to color dominating vertices, chromatic numbers and degree of vertices, are given. Particularly, we obtained necessary conditions for being extendable, and for being a Sudoku coloring. In Section 3, we determined the Sudoku number of various families of graphs. Particularly, we showed that a connected graph G has sn(G) = 1 if and only if G is bipartite. Consequently, every tree T has sn(T) = 1. We also proved that if and only if G = Kn . Moreover, a graph G with small chromatic number may have arbitrarily large Sudoku number. In Section 4, we proved that extendable partial coloring problem is NP-complete. Extendable coloring and Sudoku coloring are nice tools for providing a k-coloring of G.

Coloring, List Coloring, and Painting Squares of Graphs (and Other Related Problems)


 
 
 
 
 
 We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs. 
 
 
 
 
 
 

Adaptable and conflict colouring multigraphs with no cycles of length three or four

AbstractThe adaptable choosability of a multigraph , denoted , is the smallest integer such that any edge labelling, , of and any assignment of lists of size to the vertices of permits a list colouring, , of such that there is no edge where . Here we show that for a multigraph with maximum degree and no cycles of length 3 or 4, . Under natural restrictions we can show that the same bound holds for the conflict choosability of , which is a closely related parameter recently defined by Dvořák, Esperet, Kang and Ozeki.

The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χl(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χl(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).

Coloring Bipartite Graphs with Semi-small List Size

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex’s list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most of the parameter space. By connecting this quantity to a simple question on independent sets of hypergraphs, we strengthen bounds when a part has list size 2. Finally, we state via our framework a conjecture on general bipartite graphs, unifying three conjectures of Alon–Cambie–Kang.

Algorithmic Applications of Tree-Cut Width

The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions and provide FPT algorithms for the showcase problems Capacitated Vertex Cover, Capacitated Dominating Set, and Imbalance parameterized by the tree-cut width of an input graph. On the other hand, we show that List Coloring, Precoloring Extension, and Boolean CSP (the last parameterized by the tree-cut width of the incidence graph) are W[1]-hard and hence unlikely to be FPT when parameterized by tree-cut width.

Multiple DP-Coloring of Planar Graphs Without 3-Cycles and Normally Adjacent 4-Cycles

The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvořák and Postle (J. Combin. Theory Ser. B 129, 38–54, 2018). Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu (J. Graph Theory 93, 203–221, 2020). This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are (7m, 2m)-DP-colorable for every integer m. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most 7/2.

A sufficient condition for planar graphs to be DP-4-colorable

In 2017, Dvořák and Postle introduced DP-coloring (known as correspondence coloring) as a generalization of list coloring. Recently, a lot of attention has been put on sufficient conditions for planar graphs to be DP-4-colorable. Liu et al. [17] proved that every planar graph without triangles adjacent to cycles of length five is DP-4-colorable. Let be the family of planar graphs without 3-cycles adjacent to two cycles of length five. In this paper, we prove that every member in is DP-4-colorable, which generalizes a result of [17]. As a consequence, every graph in is 4-choosable.

Sufficient Conditions of 6-Cycles Make Planar Graphs DP-4-Colorable

In simple graphs, DP-coloring is a generalization of list coloring and thus many results of DP-coloring generalize those of list coloring. Xu and Wu proved that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. Later, Sittitrai and Nakprasit showed that if a planar graph has no pairwise adjacent 3-, 4-, and 5-cycles, then it is DP-4-colorable, which is a generalization of the result of Xu and Wu. In this paper, we extend the results on 3-, 4-, 5-, and 6-cycles by showing that every planar graph without 6-cycles simultaneously adjacent to 3-cycles, 4-cycles, and 5-cycles is DP-4-colorable, which is also a generalization of previous studies as follows: every planar graph G is DP-4-colorable if G has no 6-cycles adjacent to i-cycles where i∈{3,4,5}.

Disproof of a Conjecture by Woodall on the Choosability of $K_{s,t}$-Minor-Free Graphs

In 2001, in a survey article about list coloring, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable.&#x0D; In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon&gt;0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such that for all integers $s,t $ with $N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\varepsilon)(2s+t)$.