Decomposing a planar graph into a forest and a subgraph of restricted maximum degree.
Decomposition of complete bipartite graphs into factors with given radii
Decomposition of complete bipartite graphs into factors with given diameters
Decomposition of spheres in Hilbert spaces
Decompositions of Plane Graphs Under Parity Constrains Given by Faces
An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f. In this paper we deal with the following question: For which integers k does there exist a facial (facially proper)...
Decompositions of quadrangle-free planar graphs
W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.
Defective choosability of graphs in surfaces
It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ε, and k and d are positive integers such that k ≥ 3 and d is sufficiently large in terms of k and ε, then G is (k,d)*-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings...
Defective choosability of graphs without small minors.
Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph
In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cₘ × Kₙ, c)...
Degree sequences of -free graphs.
Degree-constrained edge partitioning in graphs arising from discrete tomography.
Determinantal generating functions of colored spanning forests.
Detour chromatic numbers
The nth detour chromatic number, χₙ(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ( G). We conjecture that χₙ(G) ≤ ⎡(τ(G))/n⎤ for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some sufficient conditions for a graph to have nth chromatic number at most 2.
Dichromatic number, circulant tournaments and Zykov sums of digraphs
The dichromatic number dc(D) of a digraph D is the smallest number of colours needed to colour the vertices of D so that no monochromatic directed cycle is created. In this paper the problem of computing the dichromatic number of a Zykov-sum of digraphs over a digraph D is reduced to that of computing a multicovering number of an hypergraph H₁(D) associated to D in a natural way. This result allows us to construct an infinite family of pairwise non isomorphic vertex-critical k-dichromatic circulant...
Die chromatischen polynome Unterringfreier Graphen.
Digraphs are 2-weight choosable.
Distance coloring of the hexagonal lattice
Motivated by the frequency assignment problem we study the d-distant coloring of the vertices of an infinite plane hexagonal lattice H. Let d be a positive integer. A d-distant coloring of the lattice H is a coloring of the vertices of H such that each pair of vertices distance at most d apart have different colors. The d-distant chromatic number of H, denoted , is the minimum number of colors needed for a d-distant coloring of H. We give the exact value of for any d odd and estimations for any...
Distance graphs from -adic norms.
Distance in stratified graphs
A graph is stratified if its vertex set is partitioned into classes, called strata. If there are strata, then is -stratified. These graphs were introduced to study problems in VLSI design. The strata in a stratified graph are also referred to as color classes. For a color in a stratified graph , the -eccentricity of a vertex of is the distance between and an -colored vertex furthest from . The minimum -eccentricity among the vertices of is the -radius of and the maximum...
Distinguishing Cartesian Products of Countable Graphs
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.