(15,36) → ¬Z;AND THUS, WE GET A CONTRADICTION AT 15

15 : - BÁO CÁO TOÁN HỌC FANS AND BUNDLES IN THE GRAPH OF PAIRWISE SUMS AND PRODUCT POTX
Toán BÁO CÁO TOÁN HỌC FANS AND BUNDLES IN THE GRAPH OF PAIRWISE SUMS AND PRODUCT POTX
15: (8,15)→ ¬x; 15: (15,14)→ ¬y; 15: (15,36) → ¬z;and thus, we get a contradiction at 15. aFinally, let us also consider the subgraph ˙G

×

+

ofG

×

+

defined as follows: The vertex-setof ˙G

×

+

is again the set of positive integers, and n joined tom if for some distinct x, y ∈Nwe have n =x+y and m =x·y.Like for G

×

+

, we can show that the chromatic number of ˙G

×

+

is at least 4, but thesubgraph which provides the counterexample is much larger.≥4.Proposition4.2. χ G˙

×

+

Proof. Let ˙G

29

be the subgraph of ˙G

×

+

induced by the 29 vertices 10, 11, 12, 13, 14, 15,