Consider a simple graph G without K 2 component with vertex set V and edge set E. Local antimagic... more Consider a simple graph G without K 2 component with vertex set V and edge set E. Local antimagic labeling f of G is a one-to-one mapping of edges to distinct positive integers 1, 2, . . . , |E| such that the weights of adjacent vertices are distinct, where the weight of a vertex is the sum of labels assigned to the edges incident to it. These weights of the vertex induced by local antimagic labeling result in a proper vertex coloring of the graph G. We define the local antimagic chromatic number of G, denoted as χ la (G), as the smallest number of distinct weights obtained across all possible local antimagic labelings of G. In this paper, we explore the local antimagic chromatic numbers of various classes of graphs, including the union of certain graph families, the corona product of graphs, and the necklace graph. Additionally, we provide constructions for infinitely many graphs for which χ la (G) equals the chromatic number χ(G) of the graph.
Let G be a graph of order n. Let f : V (G) −→ {1, 2,. .. , n} be a bijection. The weight w f (v) ... more Let G be a graph of order n. Let f : V (G) −→ {1, 2,. .. , n} be a bijection. The weight w f (v) of a vertex with respect to f is defined by w f (v) = x∈N (v) f (x). The labeling f is said to be distance antimagic if w f (u) = w f (v) for every pair of vertices u, v ∈ V (G). If the graph G admits such a labeling then G is said to be a distance antimagic graph. In this paper we investigate the existence of distance antimagic labeling in the ladder graph L n ∼ = P 2 P n .
AKCE International Journal of Graphs and Combinatorics, 2017
Let G be a graph of order n. Let f : V (G) -→ {1, 2, . . . , n} be a bijection. The weight w f (v... more Let G be a graph of order n. Let f : V (G) -→ {1, 2, . . . , n} be a bijection. The weight w f (v) of a vertex v with respect to f is defined by The labeling f is said to be distance antimagic if w f (u) ̸ = w f (v) for every pair of distinct vertices u, v ∈ V (G). If the graph G admits such a labeling, then G is said to be a distance antimagic graph. In this paper we investigate the existence of distance antimagic labelings of G + H and G • H . c
Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to... more Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to be distance antimagic if for every vertex v the vertex weight defined by $$w_fv =\sum _{x\in Nv}fx$$ is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define $$f_{k}:VG \longrightarrow \{1+k,2+k, \ldots , n+k\}$$ by $$f_{k}x=fx+k $$. If $$w_{f_{k}}u\ne \ w_{f_{k}}v$$ for every pair of vertices $$u,v \in V$$, for any $$k\ge 0$$ then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. In this paper, we provide arbitrarily distance antimagic labelings for $$rP_n$$, generalised Petersen graph $$Pn,k, \ n \ge 5$$, Harary graph $$H_{4,n}$$ for $$n\ne 6$$ and also prove that join of these graphs is distance antimagic.
Consider a simple graph G without K 2 component with vertex set V and edge set E. Local antimagic... more Consider a simple graph G without K 2 component with vertex set V and edge set E. Local antimagic labeling f of G is a one-to-one mapping of edges to distinct positive integers 1, 2, . . . , |E| such that the weights of adjacent vertices are distinct, where the weight of a vertex is the sum of labels assigned to the edges incident to it. These weights of the vertex induced by local antimagic labeling result in a proper vertex coloring of the graph G. We define the local antimagic chromatic number of G, denoted as χ la (G), as the smallest number of distinct weights obtained across all possible local antimagic labelings of G. In this paper, we explore the local antimagic chromatic numbers of various classes of graphs, including the union of certain graph families, the corona product of graphs, and the necklace graph. Additionally, we provide constructions for infinitely many graphs for which χ la (G) equals the chromatic number χ(G) of the graph.
Let G be a graph of order n. Let f : V (G) −→ {1, 2,. .. , n} be a bijection. The weight w f (v) ... more Let G be a graph of order n. Let f : V (G) −→ {1, 2,. .. , n} be a bijection. The weight w f (v) of a vertex with respect to f is defined by w f (v) = x∈N (v) f (x). The labeling f is said to be distance antimagic if w f (u) = w f (v) for every pair of vertices u, v ∈ V (G). If the graph G admits such a labeling then G is said to be a distance antimagic graph. In this paper we investigate the existence of distance antimagic labeling in the ladder graph L n ∼ = P 2 P n .
AKCE International Journal of Graphs and Combinatorics, 2017
Let G be a graph of order n. Let f : V (G) -→ {1, 2, . . . , n} be a bijection. The weight w f (v... more Let G be a graph of order n. Let f : V (G) -→ {1, 2, . . . , n} be a bijection. The weight w f (v) of a vertex v with respect to f is defined by The labeling f is said to be distance antimagic if w f (u) ̸ = w f (v) for every pair of distinct vertices u, v ∈ V (G). If the graph G admits such a labeling, then G is said to be a distance antimagic graph. In this paper we investigate the existence of distance antimagic labelings of G + H and G • H . c
Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to... more Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to be distance antimagic if for every vertex v the vertex weight defined by $$w_fv =\sum _{x\in Nv}fx$$ is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define $$f_{k}:VG \longrightarrow \{1+k,2+k, \ldots , n+k\}$$ by $$f_{k}x=fx+k $$. If $$w_{f_{k}}u\ne \ w_{f_{k}}v$$ for every pair of vertices $$u,v \in V$$, for any $$k\ge 0$$ then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. In this paper, we provide arbitrarily distance antimagic labelings for $$rP_n$$, generalised Petersen graph $$Pn,k, \ n \ge 5$$, Harary graph $$H_{4,n}$$ for $$n\ne 6$$ and also prove that join of these graphs is distance antimagic.
Uploads
Papers by ADARSH Handa