Connected graphs without long paths

On the edge spectrum of saturated graphs for paths and stars

“…Forming pendant triangles starting with ′ 2 once again gives us saturated graphs with any edge count in the interval | ( ′ 2 )| to | ( ′ 2 )| + − 3. This interval intersects the previous one obtained from 2 . Repeat this process.…”

“…In Part 1, we deal with the top part of the spectrum, in Part 2 we work on the middle part of the spectrum, and in Part 3 we give the bottom part of the spectrum. Recall that is assumed to be sufficiently large and ≥ where is defined as in (2).…”

“…In [10], Kopylov determined the extremal number for connected -free graphs on vertices. Later on, in [2], Balister et al obtained the extremal number and also gave the extremal graphs for connected -free graphs on vertices. Specifically, it was shown that the extremal graphs are of the form , , ∶= + ( −2 −1 ∪ − + +1 ), for > 2 + 1.…”

“…They also noted that the second class of extremal graph in the following theorem is of the form , ,⌊( −2)∕2⌋ = ( −2)∕2 + −( −2)∕2 when is even, and of the form , ,⌊( −2)∕2⌋ = ( −3)∕2 + ( 2 ∪ −( +1)∕2 ) when is odd. Theorem 8 [2]. Let be a connected graph on vertices containing no path on vertices, ≥ ≥ 4.…”

“…Theorem 10. Let > 0, and let and be integers with ≥ 0 ( ) and ≥ , where is defined by (2). Then for any integer such that sat( ; ) ≤ ≤ ex( ; ) − ( √ 2 + ) 3∕2 there exists asaturated graph on vertices with edges.…”