Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal {S}$ be a subset of $Z$. A vertex-coloring $\mathcal {S}$-edge-weighting of $G$ is an assignment of weight $s$ by the elements of $\mathcal {S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\{1,2\}$-edge-weighting (Lu, Yu and Zhang, (2011) \cite{LYZ}). In this paper, we show that the following result: if a 3-edge-connected bipartite graph $G$ with minimum degree $\delta$ contains a vertex $u\in V(G)$ such that $d_G(u)=\delta$ and $G-u$ is connected, then $G$ admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. In particular, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal {S}$-edge-weighting for $\mathcal {S}\in \{\{0,1\},\{1,2\}\}$. The bound is sharp, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\{1,2\}$-edge-weightings or vertex-coloring $\{0,1\}$-edge-weightings.Comment: In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring S-edge-weighting for S\in {{0,1},{1,2}

Aplikasi Dominating Set untuk Irigasi Sawah

One of the theories developed in graph theory is the dominating set. Dominating set is a concept of determining the minimum point on the graph with the provision of a point as dominating set to reach the point that is around it. The smallest cardinality of the dominating set is called the domination number denoted by (G). Given graph G with point V and E side, let D be a subset of V. If each point of VD is adjacent at least one point from D, then D is said dominating set in graph G. Currently a lot of applications from dominating set, one of which is making irrigation fields. Wetland irrigation is needed to irrigate the rice fields so that the rice field is not short of water and can irrigate sufficiently, but it takes as little as possible for irrigation making in order to irrigate the rice field well. The research focuses on application of dominating set to rice field irrigation. Keywords: dominating sets, rice field irrigatio

Aplikasi Pewarnaan Graf terhadap Penyimpanan Bahan Kimia

Graf G = (V, E), where V is the set of points and E is the set of sides. An interesting application of graphs, one of which is graph coloring. There are three kinds of coloring that are point coloring, edge coloring, and region coloring. This paper will be studied dye staining. Point coloring is coloring the dots of a graph so that no two neighboring dots have the same color. The minimum number of colors that can be used to color a graph is expressed by chromatic numbers. Currently a lot of applications for graph coloring, one of which is the storage of chemicals. The storage of chemicals required a good arrangement, this is due to the influence of chemicals on each other if stored simultaneously. The main focus of this paper is to determine the chromatic number in the graph and the application scheme of dye graph coloring. Keywords: Graph coloring, Point coloring, Edge coloring and Region coloring

Antifactors of regular bipartite graphs

Let $G=(X,Y;E)$ be a bipartite graph, where $X$ and $Y$ are color classes and $E$ is the set of edges of $G$. Lov\'asz and Plummer \cite{LoPl86} asked whether one can decide in polynomial time that a given bipartite graph $G=(X,Y; E)$ admits a 1-anti-factor, that is subset $F$ of $E$ such that $d_F(v)=1$ for all $v\in X$ and $d_F(v)\neq 1$ for all $v\in Y$. Cornu\'ejols \cite{CHP} answered this question in the affirmative. Yu and Liu \cite{YL09} asked whether, for a given integer $k\geq 3$, every $k$-regular bipartite graph contains a 1-anti-factor. This paper answers this question in the affirmative

Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$

Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$