T-coloring

Two T -colorings of a graph for T = {0, 1, 4}

In graph theory, a T-Coloring of a graph , given the set T of nonnegative integers containing 0, is a function that maps each vertex of G to a positive integer (color) such that .[1] In simple words, the absolute value of the difference between two colors of adjacent vertices must not belong to fixed set T. The concept was introduced by William K. Hale.[2] If T = {0} it reduces to common vertex coloring.

The complementary coloring of T-coloring c, denoted is defined for each vertex v of G by

where s is the largest color assigned to a vertex of G by the c function.[1]

T-chromatic number

The T-chromatic number 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. .[3]

Proof

Every T-coloring of G is also a vertex coloring of G, so . Suppose that and .
Given a common vertex k-coloring function using the colors 1, 2,..,k. We define as

For every two adjacent vertices u and w of G,


so .
Therefore d is a T-coloring of G. Since d uses k colors, .
Consequently, ■

T-span

For a T-coloring c of G, the c-span spT(c) = max {|c(u)-c(w)|} over all uw V(G).
The T-span spT(G) of G is min {spT(c)} of all colourings c of G. [4]
Some bounds of the T-span are given below:
For every k-chromatic graph G with clique of size and every finite set T of nonnegative integers containing 0, spT(K)spT(G) spT(Kk).
For every graph G and every finite set T of nonnegative integers containing 0 whose largest element is r, spT(G) ((G)-1)(r+1). .[5]
For every graph G and every finite set T of nonnegative integers containing 0 whose cardinality is t, spT(G) ((G)-1)t. .[5]

See also

References

  1. 1 2 Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. pp. 397–402.
  2. ↑ W. K. Hale, Frequency assignment: Theory and applications. Proc. IEEE 68 (1980) 1497-1514.
  3. ↑ M. B. Cozzens and F. S. Roberts, T -colorings of graphs and the Channel Assignment Problem. Congr. Numer. 35 (1982) 191-208.
  4. ↑ Chartrand, Gary; Zhang, Ping (2009). "14. Colorings, Distance, and Domination". Chromatic Graph Theory. CRC Press. p. 399.
  5. 1 2 M. B. Cozzens and F. S. Roberts, T -colorings of graphs and the Channel Assignment Problem. Congr. Numer. 35 (1982) 191-208.
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