Locating-chromatic number of the edge-amalgamation of trees

Abstract

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The investigation on the locating-chromatic number of a graph was initiated by Chartrand </span><span>et al. </span><span>(2002). This concept is in fact a special case of the partition dimension of a graph. This topic has received much attention. However, the results are still far from satisfaction. We can define the locating-chromatic number of a graph </span><span>G </span><span>as the smallest integer </span><span>k </span><span>such that there exists a </span><span>k</span><span>-partition of the vertex-set of </span><span>G </span><span>such that all vertices have distinct coordinates with respect to this partition. As we know that the metric dimension of a tree is completely solved. However, the locating-chromatic numbers for most of trees are still open. For </span><span><em>i</em> </span><span>= 1</span><span>, </span><span>2</span><span>, . . . , <em>t</em>, </span><span>let </span><em>T</em><span>i </span><span>be a tree with a fixed edge </span><span>e</span><span>o</span><span>i </span><span>called the terminal edge. The edge-amalgamation of all </span><span>T</span><span>i</span><span>s </span><span>denoted by Edge-Amal</span><span>{</span><span>T</span><span>i</span><span>;</span><span>e</span><span>o</span><span>i</span><span>} </span><span>is a tree formed by taking all the </span><span>T</span><span>i</span><span>s and identifying their terminal edges. In this paper, we study the locating-chromatic number of the edge-amalgamation of arbitrary trees. We give lower and upper bounds for their locating-chromatic numbers and show that the bounds are tight.</span></p></div></div></div>