The Neighbor-Locating-Chromatic Number of Pseudotrees

Abstract

A k-coloring of a graph G is a partition of the vertices of G into k independent sets, which are called colors. A k-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from each other by the colors of their respective neighbors. The neighbor-locating chromatic number χ N L (G) is the minimum cardinality of a neighbor-locating coloring of G.In this paper, we determine the neighbor-locating chromatic number of paths, cycles, fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring of minimum cardinality for these families of graphs is given. We also obtain tight upper bounds on the order of trees and unicyclic graphs in terms of the neighbor-locating chromatic number. Further partial results for trees are also established.

New bounds and constructions for neighbor-locating colorings of graphs

“…Thirdly, we are going to consider the case when p = q < r. If 3 = p = q < r, then let G p,q,r = C n where C n is an odd cycle of suitable length, that is, a length which will imply χ N L (C n ) = r. It is known that such a cycle exists [1,4]. As we know that χ(G p,q,r ) = 3, χ L (G p,q,r ) = 3 [8], and χ N L (G p,q,r ) = r [1,4], we are done.…”

“…and identify the vertices v 0 and u 0 . Moreover, we say that the length of the odd cycle C n is a suitable length, that is, it is of a length which ensures χ N L (C n ) = r. It is known that such a cycle exists [1,4]. Notice that χ(G p,q,r ) = p. A locating coloring f can be assigned to G p,q,r as follows:…”

“…Thus, we will need at least r colors to color C n while it is contained inside G p,q,r as a subgraph. Assign a neighbor-locating coloring c to G p,q,r as follows: assign a neighborlocating r-coloring to the odd cycle C n such that each vertex has two distinct color neighbors in case of n = k and all vertices except the two vertices say v i and v j , have two distinct color neighbors in case of n = k − 1 (refer [1] for such a neighbor-locating r-coloring). Assign distinct colors to the q − 1 leaf vertices by choosing any q − 1 colors from r colors (except c(v i ) and c(v j ) in case of n = k −1) given to the cycle C n .…”

“…Thus, we will need at least r colors to color P n while it is contained inside G p,q,r as a subgraph. Assign a neighbor-locating coloring c to G p,q,r as follows: assign a neighborlocating r-coloring to the path P n such that each vertex (except the end vertices u 0 and u n−1 ) has two distinct color-neighbors (refer [1] for such a neighbor-locating r-coloring). Choose any p − 1 distinct colors from r colors (except c(u 0 )) used in neighbor-locating r-coloring of P n and assign them to the remaining p − 1 vertices of the complete graph.…”

“…Taking cues from the concepts of location [15], neighbor-location [16] and locally identifying colorings [10], recently, two variants of graph coloring were introduced, namely, locating coloring [8] and neighbor-locating coloring [2,4]. While the former concept has been well-studied since 2002 [4,5,6,7,8,9,13,14,17,18,19]), our focus of study is the latter, which was introduced in 2014 in [4] under the name of adjacency locating coloring, renamed in 2020 in [2] and studied in a few papers since then [1,3,11,12].…”