805 research outputs found
Neighbor-locating colorings in graphs
A k -coloring of a graph G is a k -partition ¿ = { S 1 ,...,S k } of V ( G ) into independent sets, called colors . A k -coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v . The neighbor-locating chromatic number ¿ NL ( G ) is the minimum cardinality of a neighbor-locating coloring of G . We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n = 5 with neighbor-locating chromatic number n or n - 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphsPreprin
Neighbor-locating coloring: graph operations and extremal cardinalities
© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A k–coloring of a graph is a k-partition of V into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number, , is the minimum cardinality of a neighbor-locating coloring of G. In this paper, we examine the neighbor-locating chromatic number for various graph operations: the join, the disjoint union and Cartesian product. We also characterize all connected graphs of order with neighbor-locating chromatic number equal either to n or to and determine the neighbor-locating chromatic number of split graphs.Peer ReviewedPostprint (author's final draft
Characterizing All Trees with Locating-chromatic Number 3
Let be a proper -coloring of a connected graph . Let be the induced partition of by , where is the partition class having all vertices with color .The color code of vertex is the ordered-tuple , where, for .If all vertices of have distinct color codes, then iscalled a locating-coloring of .The locating-chromatic number of , denoted by , isthe smallest such that posses a locating -coloring. Clearly, any graph of order have locating-chromatic number , where . Characterizing all graphswith a certain locating-chromatic number is a difficult problem. Up to now, we have known allgraphs of order with locating chromatic number or .In this paper, we characterize all trees whose locating-chromatic number . We also give a family of trees with locating-chromatic number 4
Trees with Certain Locating-chromatic Number
The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1. They also proved that there exists a tree of order n, n≥5, having locating-chromatic number k if and only if k âˆˆ{3,4,"¦,n-2,n}. In this paper, we characterize all trees of order n with locating-chromatic number n - t, for any integers n and t, where n > t+3 and 2 ≤ t < n/2
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