Improved Bounds on the  $L(2,1)$-Number of Direct and Strong Products of Graphs
The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An <i>L</i>(2,1)-labeling of a graph <i>G</i> is a function <i>f</i> from the vertex set <i>V</i>(<i>G</i>) to the set of all nonnegative integers such that |<i>f</i>(<i>x</i>)-<i>f</i>(<i>y</i>)| ges 2 if <i>d</i>(<i>x</i>,<i>y</i>)=1 and |<i>f</i>(<i>x</i>)-<i>f</i>(<i>y</i>)| ges 1 if <i>d</i>(<i>x</i>,<i>y</i>)=2 , where <i>d</i>(<i>x</i>,<i>y</i>) denotes the distance between <i>x</i> and <i>y</i> in <i>G</i>. The <i>L</i>(2,1) -labeling number lambda(<i>G</i>) of <i>G</i> is the smallest number <i>k</i> such that <i>G</i> has an <i>L</i>(2,1)-labeling with max{<i>f</i>(<i>v</i>):<i>v</i> isin <i>V</i>(<i>G</i>)}=<i>k</i>. This paper considers the graph formed by the direct product and the strong product of two graphs and gets better bounds than those of Klavzar and Spacapan with refined approaches.