On the Radio
-chromatic Number of Paths
Abstract
A radio
-coloring of a graph
is an assignment
of positive integers (colors) to the vertices of
such that for any two vertices
and
of
, the difference between their colors is at least
. The span
of
is
. The radio
-chromatic number
of
is
. In this paper, in an attempt to prove a conjecture on the radio
-chromatic number of path, we determine the radio
-chromatic number of paths
for
if
is odd and
if
is even.
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![f](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8fa14cdd754f91cc6554c9e71929cce7.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![u](http://siba-ese.unile.it/plugins/generic/latexRender/cache/7b774effe4a349c6dd82ad4f4f21d34c.png)
![v](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9e3669d19b675bd57058fd4664205d2a.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![1+k-d(u,v)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/6fc99444a5c6b2468a7ee2276a128626.png)
![rc_k(f)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/d42f69fd93e0ecc7745f031e18b42271.png)
![f](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8fa14cdd754f91cc6554c9e71929cce7.png)
![\max\{f(v):v\in V(G)\}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/9fe8e7ab8b5222f20c5ba24943487c59.png)
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
![rc_k(G)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/52dc3004c9e62539525e691b44358471.png)
![G](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![min\lbrace rc_k(f) : f { is a radio k\text{-}coloring of } G\rbrace](http://siba-ese.unile.it/plugins/generic/latexRender/cache/ce55de242df9611cb529c64bafa7bb49.png)
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
![P_n](http://siba-ese.unile.it/plugins/generic/latexRender/cache/27e29960e16256defede6f6e0697ccf7.png)
![k+5\leq n\leq\frac{7k-1}{2}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1923f6ca6962ac230c5a3a77b3068a74.png)
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
![k+4\leq n\leq\frac{5k+4}{2}](http://siba-ese.unile.it/plugins/generic/latexRender/cache/36579b1462064ae3974432e6711340a4.png)
![k](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8ce4b16b22b58894aa86c421e8759df3.png)
DOI Code:
10.1285/i15900932v42n1p37
Keywords:
radio k-coloring; radio k-chromatic number; radio coloring; radio number
Full Text: PDF