1, 2, 3 conjecture is well-known:

If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers of edges incident with it then we obtain a proper vertex coloring.

Question: Is the following weaker statement true?

For any simple graph $G$ there are three natural numbers $p_G, q_G, r_G$ such that if :

(a) We label any edge of $G$ with a number among $p_G, q_G, r_G$.

(b) We label any vertex of $G$ with the sum of numbers of the incident edges.

Then the vertex labels form a proper vertex coloring of the $G$.

Remark: In other words the question is about truth of $1, 2, 3$ conjecture when we replace global numbers $1, 2, 3$ with numbers localized to each given $G$.

The 1, 2, 3 conjecture is well-known:

If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers of edges incident with it then we obtain a proper vertex coloring.

Question: Is the following weaker statement true?

For any simple graph $G$ there are three natural numbers $p_G, q_G, r_G$ such that if :

(a) We label any edge of $G$ with a number among $p_G, q_G, r_G$.

(b) We label any vertex of $G$ with the sum of numbers of the incident edges.

Then the vertex labels form a proper vertex coloring of the $G$.

Remark: In other words the question is about the truth of the $1, 2, 3$ conjecture when we replace global numbers $1, 2, 3$ with numbers localized to each given $G$.

1, 2, 3 conjecture is well-known:

If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers of edges incident with it then we obtain a vertex proper coloring.

Question: Is the following weaker statement true?

For any simple graph $G$ there are three natural numbers $p_G, q_G, r_G$ such that if :

(a) We label any edge of $G$ with a number among $p_G, q_G, r_G$.

(b) We label any vertex of $G$ with the sum of numbers of the incident edges.

Then the number of vertexes forms a proper vertex coloring of the $G$.

Remark: In the other words the question is about truth of $1, 2, 3$ conjecture when we replace global numbers $1, 2, 3$ with numbers localized to each given $G$.

1, 2, 3 conjecture is well-known:

If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers of edges incident with it then we obtain a proper vertex coloring.

Question: Is the following weaker statement true?

For any simple graph $G$ there are three natural numbers $p_G, q_G, r_G$ such that if :

(a) We label any edge of $G$ with a number among $p_G, q_G, r_G$.

(b) We label any vertex of $G$ with the sum of numbers of the incident edges.

Then the vertex labels form a proper vertex coloring of the $G$.

Remark: In other words the question is about truth of $1, 2, 3$ conjecture when we replace global numbers $1, 2, 3$ with numbers localized to each given $G$.