The chromatic gap and its extremes

The chromatic gap is the difference between the chromatic number and the clique number of a graph. Here we investigate gap(n), the maximum chromatic gap over graphs on n vertices. Can the extremal graphs be explored? While computational problems related to the chromatic gap are hopeless, an interplay between Ramsey theory and matching theory leads to a simple and (almost) exact formula for gap(n) in terms of Ramsey numbers. For our purposes it is more convenient to work with the covering gap, the difference between the clique cover number and stability number of a graph and this is what we call the gap of a graph. Then gap(n) can be equivalently defined (by switching from a graph to its complement), as the maximum gap over graphs of n vertices. Notice that the well-studied family of perfect graphs are the graphs whose induced subgraphs have gap zero. Our study is a first step towards better understanding of graphs whose induced subgraphs have gap at most t. The maximum of the (covering) gap and the chromatic gap running on all induced subgraphs will be called perfectness gap. Using α(G) for the cardinality of a largest stable (independent) set of a graph G, we define α(n) = minα(G) where the minimum is taken over trianglefree graphs on n vertices. It is easy to observe that α(n) is essentially an inverse Ramsey function, defined by the relation R(3, α(n)) ≤ n < R(3, α(n)+1). Our main result is that gap(n) = ⌈n/2⌉−α(n), possibly with the exception of small intervals (of length at most 15) around the Ramsey numbers R(3,m), where the error is at most 3. The central notions in our investigations are the gap-critical and the gapextremal graphs. A graph G is gap-critical if for every proper induced subgraph H ⊂ G, gap(H) < gap(G) and gap-extremal if it is gap-critical with as few vertices as possible (among gap-critical graphs of the same gap). The strong perfect graph theorem, solving a long standing conjecture of Berge that stimulated a broad area of research, states that gap-critical graphs with gap 1 are the holes (chordless odd cycles of length at least five) and antiholes (complements of holes). The next step, the complete description of gap-critical graphs with gap 2 would probably be a very difficult task. As a very first step, we prove that there is a unique 2-extremal graph, 2C5, the union of two disjoint (chordless)