The chromatic gap and its extremes

• Authors: AndráS GyáRfáS;AndráS Seb;Nicolas Trotignon
• Venue: Journal of Combinatorial Theory Series B
• Year: 2012

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. Notice that the well-studied family of perfect graphs are the graphs whose induced subgraphs have gap zero. The maximum of the (covering) gap and the chromatic gap running on all induced subgraphs will be called perfectness gap. Using @a(G) for the cardinality of a largest stable (independent) set of a graph G, we define @a(n)=min@a(G) where the minimum is taken over triangle-free graphs on n vertices. It is easy to observe that @a(n) is essentially an inverse Ramsey function, defined by the relation R(3,@a(n))==0, we denote by s(t) the smallest order of a graph with gap t and we call a graph is t-extremal if it has gap t and order s(t). Equivalently, s(t) is the smallest order of a graph with perfectness gap equal to t. It is tempting to conjecture that s(t)=5t with equality for the graph tC"5. However, for t=3 the graph tC"5 has gap t but it is not gap-extremal (although gap-critical). We shall prove that s(3)=13, s(4)=17 and s(5)@?{20,21}. Somewhat surprisingly, after the uncertain values s(6)@?{23,24,25}, s(7)@?{26,27,28}, s(8)@?{29,30,31}, s(9)@?{32,33} we can show that s(10)=35. On the other hand we can easily show that s(t) is asymptotically equal to 2t, that is, gap(n) is asymptotic to n/2. According to our main result the gap is actually equal to @?n/2@?-@a(n), unless n is in an interval [R,R+14] where R is a Ramsey-number, and if this exception occurs the gap may be larger than this value by only a small constant (at most 3). Our study provides some new properties of Ramsey graphs them selves: it shows that triangle-free Ramsey graphs have high matchability and connectivity properties, leading possibly to new bounds on Ramsey-numbers.