Some new upper bounds of ex(n;{C3,C4})

Graphs with girth at least 5 with orders between 20 and 32

“…Suppose that there are two non-intersecting blocks, then they must each have weight at most 10 and each contain a pair of vertices of regularity at most 2 (from Lemma 7). It follows that (deg 2 , deg 3 ) = (4,8). But then two of the blocks contain only points of regularity 3 and have weight 12 which contradicts Lemma 7.…”

“…In [9] and [15] algorithms are applied to compute lower bounds on f 4 (v) for 31 ≤ v ≤ 200. Some of these lower bounds are improved in [1] and improved upper bounds for 33 ≤ v ≤ 42 are proved in [4]. Values of f 4 (v) for v ≤ 30, and of F 4 (v) for v ≤ 21 are available as sequences A006856 and A159847 of the On-Line Encyclopedia of Integer Sequences [13].…”

“…For any extremal graph, δ and ∆ denote the minimum and maximum degree respectively. v f (v) (δ, ∆) 21 44 (3,5), (3,6), (4, 5) 22 47 (3,5), (3,6), (3,7), (4, 5) 23 50 (3,5), (3,6), (3,7), (4, 5) 24 54 (4, 5) 25 57 (3,5), (3,6), (3,7), (3,8), (4,5), (4, 6) 26 61 (4,5), (4, 6) 27 65 (4,5), (4, 6) 28 68 (3,5), (3,6), (3,7), (3,8), (3,9), (4,5), (4, 6) 29 72 (4,5), (4,6), (4, 7) 30 76 (4,6), (4, 7) 31 80 (4,6),…”

“…Note that S, H denotes that some of the graphs were found using search, and some by hand. v f (v) (δ, ∆) Embedded star F (v) method 20 41 (3, 5) S 5, [3,3,3,3,2] 1 G 21 44 (3, 5) S 5, [3,3,3,3,3] 3 H (4, 5) S 5, [3,3,3,3,3] 22 47 (3, 5) S 5, [4,3,3,3,3] 3 S (4, 5) S 5, [4,3,3,3,3] 23 50 (3, 5) S 5, [4,4,3,3,3] 7 S (4, 5) S 5, [4,4,3,3,3] or S 5, [4,3,3,3,3] 24 54 (4, 5) S 5, [4,4,4,3,3] 1 S 25 57 (3, 5) S 5, [4,4,4,4,3] or S 5, [4,…”

“…Note that, since all the other points in the blocks containing p must be distinct, if all blocks have size r, then reg p ≤ (v − 1)/(r − 1). v P (3, v) P (4, v) P (5, v) P (6, v) 4 (ii) If we sum the weights of the blocks, for every vertex p, reg p is counted reg p times. So Σ b∈B wt(b) = Σ p (reg 2 p ).…”