Search

The graphs used in this paper are simple, connected and finite graph. A graceful m -coloring of a graph G is a proper vertex coloring c : V (G) → { 1 , 2 , m } , where m ≥ 2 which induces a proper edge coloring c ′ : E (G) → { 1 , 2 , m − 1 } that defined by c ′ (u v) = | c (u) − c (v) |. Proper vertex coloring c of a graph G is a graceful coloring if c is a graceful m-coloring for m ∈ N. The minimum vertex coloring, that can apply such that a graph G can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by χ g (G). In this paper, we will investigate the graceful chromatic number of generalized Petersen graph P (n , k) for k = 1 , 2 , 3 , 4. [ABSTRACT FROM AUTHOR]

Tuberculosis (TB) is an infectious disease with a high incidence rate in Indonesia. This disease is a health problem nationally and globally. Disobedience of TB patients in taking medication is an obstacle in healing. PMO (Drug Supervisor) officers have a very important duty to ensure TB patients are disciplined in taking their medication, so they can improve patient's quality of life. However, majority of patients forget to take medicine frequently and PMO to remind them in taking medicine unconsistently. This means that there is a high probability that good behavior that is carried out after education is not consistent, because patients do not have adequate support systems to comply with treatment and orderly in preventing transmission. The purpose of this study was to design an android-based educational application for TB patients in Tasikmalaya city. Prototype design was used in this research. The results found 4 features in the application design such as education (providing health education through pictures, posters and videos), screening (tracking of families are at high risk of spreading), taking medication (remind and evaluate of taking medication) and consultation (counseling related to disease). Based on the test results, it can be concluded that the SMART-TB application design can able to provide information and convenience for some TB patients in reporting taking medication and health-related consultations. [ABSTRACT FROM AUTHOR]

We introduce the notion of distance irregular labeling, called the local distance ir-regular labeling. We define λ: V (G) → {1,2;... k} kg such that the weight calculated at the vertices induces a vertex coloring if w(u) 6≠ w(v) for any edge uv. The weight of a vertex u ∈ V (G) is defined as the sum of the labels of all vertices adjacent to u (distance 1 from u), that is w(u) = ∑y∈N(u)λ(y). The minimum cardinality of the largest label over all such irregular assignment is called the local distance irregularity strength, denoted by disl(G). In this paper, we found the lower bound of the local distance irregularity strength of graphs G and also exact values of some classes of graphs namely path, cycle, star graph, complete graph, (n;m)-tadpole graph, unicycle with two pendant, binary tree graph, complete bipartite graphs, sun graph. [ABSTRACT FROM AUTHOR]

A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,...,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,...,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,...,Gm, and then joining every vertex u in Gi, i∈{1,2,...,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles. [ABSTRACT FROM AUTHOR]

Background : One of the topics of distance in graphs is the resolving set problem. Suppose the set W = { s 1 , s 2 , …, s k } ⊂ V ( G ), the vertex representations of ∈ V ( G ) is r m ( x | W ) = { d ( x , s 1 ), d ( x , s 2 ), …, d ( x , s k )}, where d ( x , s i ) is the length of the shortest path of the vertex x and the vertex in W together with their multiplicity. The set W is called a local m -resolving set of graphs G if r m ( v | W )≠ r m ( u | W ) for uv ∈ E ( G ). The local m -resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of G , denoted by md l ( G ). Thus, if G has an infinite local multiset dimension and then we write md l G = ∞ . Methods : This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set W ⊂ V ( G ); determine the vertex representation of two adjacent vertices in G ; and prove the theorem. Results : The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely md l ( Amal ( G , v , m )) ≤ m. md l ( G ) and their exact value of local multiset dimension of some families of graphs namely md l ( Amal ( P n , v , m )) = 1, md l Amal K n v m = ∞ , md l ( Amal ( W n , v , m )) = m. md l ( W n ), md l ( Amal ( F n , v , m )) = m. md l ( F n ) for d ( v ) = n , md l Amal F n v m = m . n 4 . Conclusions : We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs., Competing Interests: No competing interests were disclosed., (Copyright: © 2024 Alfarisi R et al.)