**Conjecture**There is an absolute constant such that for every hexagonal graph and vertex weighting ,

A *hexagonal graph* is an induced subgraph of the triangular lattice. The *triangular lattice* may be described as follows. The vertices are all integer linear combinations of the two vectors and . Two vertices are adjacent when the Euclidean distance between them is 1.

Let be a graph and a vertex weighting . The * weighted clique number* of , denoted by , is the maximum weight of a clique, that is , where . A *-colouring* of a is a mapping such that for every vertex , and for all edge , . The *chromatic number* of , denoted by , is the least integer such that admits a -colouring.

The conjecture would be tight because of the cycle of length 9. The maximum size of stable set in is . Thus and , where is the all function.

McDiarmid and Reed [MR] proved that for any hexagonal graph and vertex weighting . Havet [H] proved that if a hexagonal graph is triangle-free, then (See also [SV]).

The conjecture would be implied by the following one, where is the all function.

**Conjecture**for every hexagonal graph.

Since , where is the stability number (the maximum size of a stable set). A first step to this later conjecture would be to prove the following conjecture of McDiarmid.

**Conjecture**Let be a triangle-free hexagonal graph.

## Bibliography

[H] F.Havet. Channel assignment and multicolouring of the induced subgraphs of the triangular lattice. Discrete Mathematics 233:219--231, 2001.

*[MR] C. McDiarmid and B. Reed. Channel assignment and weighted coloring, Networks, 36:114--117, 2000.

[SV] K. S. Sudeep and S. Vishwanathan. A technique for multicoloring triangle-free hexagonal graphs. Discrete Mathematics, 300(1-3), 256--259, 2005.

* indicates original appearance(s) of problem.