PEWARNAAN TITIK TOTAL SUPER ANTI-AJAIB LOKAL PADA GRAF PETERSEN DIPERUMUM P(n,k) DENGAN k=1,2

The local antimagic total vertex labeling of graph G is a labeling that every vertices and edges label by natural number from 1 to such that every two adjacent vertices has different weights, where is The sum of a vertex label and the labels of all edges that incident to the vertex. If the labeling start the smallest label from the vertex then the edge so that kind of coloring is called the local super antimagic total vertex labeling. That local super antimagic total vertex labeling induces vertex coloring of graph G where for vertex v, the weight w(v) is the color of v. The minimum number of colors that obtained by coloring that induces by local super antimagic total vertex labeling of G called the chromatic number of local super antimagic total vertex coloring of G, denoted by χlsat(G). In this paper, we consider the chromatic number of local super antimagic total vertex coloring of Generalized Petersen Graph P(n,k) for k=1, 2.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

Reducing Hamiltonian path to a vertex label

Given a graph 𝑮 = (𝑽,𝑬), we want to label all the vertices 𝑣 ∈ 𝑽 with values from {1, 2, … , 𝑛} where |𝑽| = 𝑛 such that for all edges (𝑥, 𝑦) ∈ 𝑬 such that |𝑙𝑎𝑏𝑒𝑙(𝑥) − 𝑙𝑎𝑏𝑒𝑙(𝑦)| ≥ 2. We have to determine whether such a labelling is possible for a graph.

Natural Scene Recognition Based on Graph Edit Distance

Natural scene classification is a challenging pattern classification problem nowadays. The description of image plays a crucial role in the process of recognition. Many different approaches and feature extraction methodologies concerning scene classification have been proposed and applied in the last few years. This paper proposed a novel method of natural scene recognition based on graph edit distance (GED) in which scene images are represented by attributed graph. The vertex label is the features of regions and edge label is the features of public area of adjacent regions. This method used local representation as well as global way, realized the cooperation of global and local mechanisms. The proposed method approaches satisfactory categorization performances on the well-known scene classification datasets with 8 scene categories.

A total labellings of m triangles

Assume that we have $m$ triangles. In this paper, we discuss certain labelling of the $m$ triangles called $c$-Erd\"osian for some positive integers $c$. We regard labellings of the vertices of the triangles by positive integers, which induce the edge labels for the triangles as the sum of the two incident vertex labels. They have the property that each vertex label and edge label appears only once in the set of positive integers $\{c,\ldots ,c+6m-1\}$. Here, we show how to construct certain $c$-Erd\"osian of $m$ triangles.