Margit Voigt

List colourings of planar graphs

A graph G=G(V,E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erdos, Rubin and Taylor 1979 about the choosability of planar graphs:(1)every planar graph is 5-choosable and, (2)there are planar graphs which are not 4-choosable. We will prove the second conjecture.

On the b-Chromatic Number of Graphs

The b-chromatic number b(G) of a graph G = (V, E) is the largest integer k such that G admits a vertex partition into k independent sets Xi (i = 1, . . . , k) such that each Xi contains a vertex xi adjacent to at least one vertex of each Xj, j ? i. We discuss on the tightness of some bounds on b(G) and on the complexity of determining b(G). We also determine the asymptotic behavior of b(Gn, p) for the random graph, within the accuracy of a multiplicative factor 2 + o(1) as n ? ?.

A not 3-choosable planar graph without 3-cycles

Abstract An L - list coloring of a graph G is a proper vertex coloring in which every vertex v receives a color from a prescribed list L ( v ). G is called k - choosable if all lists L ( v ) have the cardinality k and G is L -list colorable for all possible assignments of such lists. Recently, Thomassen has proved that every planar graph with girth greater than 4 is 3-choosable. Furthermore, it is known that the chromatic number of a planar graph without 3-cycles is at most 3. Consequently, the question resulted whether every planar graph without 3-cycles is 3-choosable. In the following we will give a planar graph without 3-cycles which is not 3-choosable.

On Dominating Sets and Independent Sets of Graphs

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 ≤ ki ≤ di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ VsDk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) ∣ pi ∈ R, 0 ≤ pi ≤ 1, i = 1, …, n}. An O(Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with ∣Dk∣≤fk(p).

Chromatic number of prime distance graphs

For any set D of positive integers, the distance graph G(D)G(V,E) is the graph with vertex set V(G)Z and edge set E(G){(u,v):|u−v|∈D}. In Research Problem 77 (Discrete Math. 69 (1988) 105–106) Eggleton, Erdos and Skilton propose the problem to determine all minimal subsets D of the prime numbers such that graph G(D) is 4-chromatic. In the present paper this problem is solved for 4-element prime sets D.

Distance Graphs with Finite Chromatic Number

The distance graph G(D) with distance set D={d1, d2, ?} has the set Z of integers as vertex set, with two vertices i, j?Z adjacent if and only if |i?j|?D. We prove that the chromatic number of G(D) is finite whenever inf{di+1/di}>1 and that every growth speed smaller than this admits a distance set D with infinite-chromatic G(D).

Minimal colorings for properly colored subgraphs

We give conditions on the minimum numberk of colors, sufficient for the existence of given types of properly edge-colored subgraphs in ak-edge-colored complete graph. The types of subgraphs we study include families of internally pairwise vertex-disjoint paths with common endpoints, hamiltonian paths and hamiltonian cycles, cycles with a given lower bound of their length, spanning trees, stars, and cliques. Throughout the paper, related conjectures are proposed.

Choosability and fractional chromatic numbers

Abstract A graph G is ( a , b )-choosable if for any assignment of a list of a colors to each of its vertices there is a subset of b colors of each list so that subsets corresponding to adjacent vertices are disjoint. It is shown that for every graph G , the minimum ratio a b where a , b range over all pairs of integers for which G is ( a , b )-choosable is equal to the fractional chromatic number of G .

On 3-colorable non-4-choosable planar graphs

An L-list coloring of a graph G is a proper vertex coloring in which every vertex v gets a color from a list L(v) of allowed colors. G is called k-choosable if all lists L(v) have exactly k elements and if G is L-list colorable for all possible assignments of such lists. Verifying conjectures of Erdos, Rubin and Taylor it was shown during the last years that every planar graph is 5-choosable and that there are planar graphs which are not 4-choosable. The question whether there are 3-colorable planar graphs which are not 4-choosable remained unsolved. The smallest known example far a non-4-choosable planar graph has 75 vertices and is described by Gutner. In fact, this graph is also 3 colorable and answers the above question. In addition, we give a list assignment for this graph using 5 colors only in all of the lists together such that the graph is not List-colorable.

A non-3-choosable planar graph without cycles of length 4 and 5

Steinbergs question from 1975 whether every planar graph without 4- and 5-cycles is 3-colorable is still open. In this paper the analogous question for 3-choosability of such graphs is answered to the negative.