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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)