The bandwidth of a graph is the minimum, over vertex labelings with distinct integers, of the maximum difference between labels on adjacent vertices. Kuang and McDiarmid proved that almost all n-vertex graphs have bandwidth n - (2 + √2 + o(1))log2 n. Thus the sum of the bandwidths of a graph and its complement is almost always at least 2n - (4 + 2√2 + o(1))log2 n; we prove that it is always at most 2n - 4log2 n + o(log n). The proofs involve improving the bounds on the Ramsey and Turán numbers of the "halfgraph".
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics