Locating-Chromatic Number of Amalgamation of Stars

Let G be a connected graph and c a proper coloring of G . For i Æ’­1,2,Æ’»,k define the color class i C as the set of vertices receiving color i . The color code c (v) "ž¨ of a vertex v in G is the ordered k -tuple 1 ( ( , ), , ( , )) k d v C Æ’» d v C where ( , ) i d v C is the distance of v to i C . If all distinct vertices of G have distinct color codes, then c is called a locating-coloring of G . The locating-chromatic number of graph G , denoted by ( ) L ƒÓ G is the smallest k such that G has a locating coloring with k colors. In this paper we discuss the locating-chromatic number of amalgamation of stars k ,m S . k ,m S is obtained from k copies of star 1,m K by identifying a leaf from each star. We also determine a sufficient condition for a connected subgraph k ,m H "ž~ S satisfying , ( ) ( ) L L k m ƒÓ H "žT ƒÓ

High degree graphs contain large-star factors

We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $\Omega((d/\log d)^{1/3})$. This settles a problem of J. Kratochvil