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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)

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)

Let Q n 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… (More)

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P, Q)-total coloring of a simple graph G is a coloring of the vertices V (G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of… (More)

Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [Z r ] s be the set of all s-element subsets of Z r. An (r, s)-fractional (P, Q)-total coloring of G is an assignment h : V (G) ∪ E(G) → [Z r ] s such that for each i ∈ Z r the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property… (More)