A path v1,v2,…,vm in a graph G is degree-monotone if deg(v1)≤deg(v2)≤⋯≤deg(vm) where deg(vi) is the degree of vi in G. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by Mk(m) the minimum number M such that for all n≥M, in any k-edge coloring of Kn there is some 1≤j≤k such that the graph formed by the edges colored j has a degree-monotone path of order m. We prove several nontrivial upper and lower bounds for Mk(m).
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© 2016 Elsevier B.V.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics