T coloring

T-chromatic number

The T-chromatic number χ T ( G ) is the minimum number of colors that can be used in a T-coloring of G. The T-chromatic number is equal to the chromatic number. χ T ( G ) = χ ( G ) .

Proof

Every T-coloring of G is also a vertex coloring of G, so χ T ( G ) ≥ χ ( G ) . Suppose that χ ( G ) = k and r = m a x ( T ) .
Given a common vertex k-coloring function c : V ( G ) → N using the colors 1, 2,..,k. We define d : V ( G ) → N as
d ( v ) = ( r + 1 ) c ( v )
For every two adjacent vertices u and w of G,
| d ( u ) − d ( w ) | = | ( r + 1 ) c ( u ) − ( r + 1 ) c ( w ) |
= ( r + 1 ) | c ( u ) − c ( w ) | ≥ r + 1
so | d ( u ) − d ( w ) | ∉ T .
Therefore d is a T-coloring of G. Since d uses k colors, χ T ( G ) ≤ k = χ ( G ) .
Consequently, χ T ( G ) = χ ( G ) ■

T-span

For a T-coloring c of G, the c-span sp_T(c) = max {|c(u)-c(w)|} over all uw ∈ V(G).
The T-span sp_T(G) of G is min {sp_T(c)} of all colourings c of G.
Some bounds of the T-span are given below:
For every k-chromatic graph G with clique of size ω and every finite set T of nonnegative integers containing 0, sp_T(K _ω ) ≤ sp_T(G) ≤ sp_T(K_k).
For every graph G and every finite set T of nonnegative integers containing 0 whose largest element is r, sp_T(G) ≤ ( χ (G)-1)(r+1). χ T ( G ) = χ ( G ) .
For every graph G and every finite set T of nonnegative integers containing 0 whose cardinality is t, sp_T(G) ≤ ( χ (G)-1)t. χ T ( G ) = χ ( G ) .