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

DIMENSI PARTISI DAN DIMENSI PARTISI BINTANG GRAF HASIL OPERASI COMB DUA GRAF TERHUBUNG

Misalkan $G$ adalah sebuah graf nontrivial dan terhubung dengan himpunan simpul $V(G)$, himpunan sisi $E(G)$ dan $S\subseteq V(G)$ dengan simpul $v\in V(G)$, jarak antara $v$ dan $S$ adalah $d(v,S)=$min$\{d(v,x)|x\in S\}$. Untuk sebuah partisi $\Pi=\{S_1,S_2,S_3,...,S_k\}$ dari $V(G)$, representasi simpul $v$ terhadap $\Pi$ didefinisikan oleh pasangan $r(v|\Pi)=(d(v,S_1),d(v,S_2),...,d(v,S_k))$. Partisi $\Pi$ disebut partisi pembeda dari $G$ jika semua representasi dari setiap simpul $v\in V(G)$ berdeda. Kardinalitas minimum dari partisi pembeda disebut dimensi partisi dari $G$ dan dinotasikan sebagai pd($G$). Ragam lain dari konsep dimensi partisi yaitu dimensi partisi bintang. Misalkan $\Pi=\{S_1,S_2,S_3,...,S_k\}$ disebut partisi pembeda bintang jika setiap kelas-kelas partisi $S_i, 1\leq i\leq k$ menginduksi sebuah graf bintang di $G$ dan semua representasi dari setiap simpul $v\in V(G)$ berdeda. Kardinalitas minimum dari partisi pembeda bintang disebut dimensi partisi bintang dari $G$ dan dinotasikan sebagai spd($G$). Dalam Penelitian ini, kami akan menentukan dimensi partisi dan dimensi partisi bintang dari graf hasil operasi produk $comb$. Operasi $comb$ dinotasikan $\triangleright$. Untuk graf $G$ dan $H$, graf hasil operasi $comb$ $G\triangleright H$ didefinisikan sebagai graf yang diperoleh dengan mengambil satu duplikat $G$ dan $|G|$ duplikat dari $H$ dan melekatkan simpul $u$ dari masing masing graf $H$ duplikat ke-$i$ pada simpul ke-$i$ dari graf $G$. Misalkan $G$ dan $H$ adalah graf terhubung meliputi lintasan, lingkaran, dan graf lengkap. ================================================================= Let $G$ be a nontrivial and connected graphs with vertex set $V(G)$, edge set $E(G)$ and $S\subseteq V(G)$ with vertex $v\in V(G)$. The distance between $v$ and $S$ is $d(v,S)=$min$\{d(v,x)\}$ for $x\in S$. For an ordered partition $\Pi=\{S_1,S_2,S_3,...,S_k\}$ of vertex set $V(G)$, the representation of $v$ with respect to $\Pi$ is defined by the ordered $r(v|\Pi)=(d(v,S_1),d(v,S_2),...,d(v,S_k))$. The minimum cardinality of resolving partition is partition dimension of $G$, denoted by pd($G$). A variant of partition dimension concept called star partition dimension of a graph. Let $\Pi = \{S_1,S_2,S_3,...,S_k\}$ be a star resolving partition for $G$ if each partition class $S_i, \,1\leq i\leq k$, induces a star in $G$ and all representation of vertices $v\in V(G)$ are unique. The minimum cardinality of resolving partition is a star partition dimension of $G$, denoted by spd($G$). In this research, we determine the partition dimension and star partition dimension of comb product of graphs. For graphs $G$ and $H$, the comb product $G\triangleright H$ is defined as the graph obtained by taking one copy of $G$ and $|V(G)|$ copies of $H$ and grafting the $i$-th copy of $H$ at the vertex $o$ to the $i$-th vertex of $G$. In this work $G$ and $H$ are restricted to path, cycle and complete graph