# Arnfried Kemnitz

• Discrete Mathematics
• 1998
An integer distance graph is a graph G(D) with the set of integers as vertex set and with an edge joining two vertices u and v if and only if ju ? vj 2 D where D is a subset of the positive integers. We determine the chromatic number (D) of G(D) for some nite distance sets D such as sets of consecutive integers and special sets of cardinality 4.
• Discrete Mathematics
• 2001
A plane integral drawing of a planar graph G is a realization of G in the plane such that the vertices of G are mapped into distinct points and the edges of G are mapped into straight line segments of integer length which connect the corresponding vertices such that two edges have no inner point in common. We conjecture that plane integral drawings exist(More)
Given positive integers k and d with k ≥ 2d, a (k, d)-total coloring of a simple and finite graph G is an assignment c of colors {0, 1, . . . , k−1} to the vertices and edges of G such that d ≤ |c(x)− c(x)| ≤ k − d whenever x and x are two adjacent edges, two adjacent vertices or an edge incident to a vertex. The circular total chromatic number χ c (G) is(More)
• Graphs and Combinatorics
• 2007
Let Qn be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G, H) be the largest number of colors such that there exists an edge coloring of(More)
• Discussiones Mathematicae Graph Theory
• 2015
Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [Zr] be the set of all s-element subsets of Zr. An (r, s)-fractional This work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by the Slovak VEGA Grant 1/0652/12. 2 A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R.(More)
• Discussiones Mathematicae Graph Theory
• 2011
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that rc(G) = 2 for every connected graph G of order n(More)
• Discrete Mathematics
• 2001
An integer distance graph is a graph G(D) with the set of integers as vertex set and with an edge joining two vertices u and v if and only if ju ? vj 2 D where D is a subset of the positive integers. We determine the chromatic number (D) of G(D) if D is a 4-element set of the form