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The Turán number of a graph H , denoted by e x ( n , H ) , is the maximum number of edges in a simple graph of order n which does not contain H as a subgraph. In this paper, we determine the value e x ( n , k ⋅ P 3 ) and characterize all extremal graphs for all positive integers n and k , where k ⋅ P 3 is k disjoint copies of a path on three vertices. This extends...
Given a graph F and an integer r ≥ 2 , a partition F ̂ of the edge set of F into at most r classes, and a graph G , define c r , F ̂ ( G ) as the number of r -colorings of the edges of G that do not contain a copy of F such that the edge partition induced by the coloring is isomorphic to the one of F . We think of F ̂ as the pattern of coloring that should...
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