Let k=3 or 4, and let n be a natural number not divisible by k-1. Consider any edge coloring of the complete graph K of order (k-1)(n-1)+2 with k colors. The following facts were known previously: 1. (i) K contains a monochromatic connected subgraph on more than n vertices. 2. (ii) There are k-1 monochromatic connected subgraphs whose union covers the entire vertex set of K. We prove that the requirements of (i) and (ii) can be fulfilled simultaneously, i.e. 3. (iii) There are k-1 monochromatic connected subgraphs G1,..., Gk-1 such that |V(G1)|≥n+1 and V(G1)∩⋯∩V(Gk-1)=V(K).
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics