Locally Dense Independent Sets in Regular Graphs of Large Girth—An Example of a New Approach

We present an example for a new approach which seems applicable to every graph theoretical concept defined by local conditions and regular graphs of large girth. It combines a random outer procedure processing the graph in rounds with a virtually arbitrary algorithm solving local instances within each round and combines the local solutions to a global one. The local uniformity of the considered instances and the randomness of the outer procedure make the asymptotic analysis possible. Here we apply this approach to the simplest yet fundamental example of a locally defined graph theoretical concept: independent sets in graphs.

For an integer

d

≥3 let

α

(

d

) be the supremum over all

α

with the property that for every

ε

>0 there exists some

g

(

ε

) such that every

d

-regular graph of order

n

and girth at least

g

(

ε

) has an independent set of cardinality at least (

α

−

ε

)

n

.

Considerably extending the work of Lauer and Wormald (Large independent sets in regular graphs of large girth, J. Comb. Theory, Ser. B

97

, 999–1009, 2007) and improving results due to Shearer (A note on the independence number of triangle-free graphs, II, J. Comb. Theory, Ser. B

53

, 300–307, 1991) and Lauer and Wormald, we present the best known lower bounds for

α

(

d

) for all

d

≥3.