Complete subgraphs are elusive


We would like to decide whether a graph with a given vertex set has a certain property. We do this by probing the pairs of vertices one by one, i.e., asking whether a given pair of vertices is an edge or not. At each stage we make full use of the information we have up to that point. If there is an algorithm (a sequence of probes depending on the previous information) that allows us to come to a decision before checking every pair, the property is said to be incomplete, otherwise it is called complete or elusive. We show that the property of containing a complete subgraph with a given number of vertices is elusive. The proof also implies that the property of being r-chromatic (r fixed) is elusive. © 1976.

Publication Title

Journal of Combinatorial Theory, Series B