Local antimagic vertex coloring of unicyclic graphs

The local antimagic labeling on a graph G with |V | vertices and |E| edges is defined to be an assignment f : E → {1, 2, · · · , |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u) 6= w(v) where w(u) = Σe∈E(u)f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.


Introduction
Let G = (V, E) be a finite, simple, connected and undirected graph.The local antimagic labeling on a graph G with |V | vertices and |E| edges is defined to an assignment f : E → {1, 2, • • • , m} so that the weights any two adjacent vertices are distinct, that is, w(u) = w(v) where w(u) = Σ e∈E(u) f (e) and E(u) is the set of edges incident to u.Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u).The local antimagic chromatic number, denoted by χ la (G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G.This concept was recently introduced by Arumugam et al. [1].
In the paper [1], Arumugam et al. presented the exact value of the local antimagic chromatic number of some families of graphs as follows.
• Friendship graph F n for n ≥ 2 by removing an edge e, χ la (F n − {e}) = 3.
• Graph L n for n ≥ 2 that is obtained by inserting a vertex to each edge vv i , 1 Furthermore, Arumugam et al. [1] showed that for any tree T with l leaves, χ la (T ) ≥ l + 1 and for the graph H = G + K2 where G is a graph of order n ≥ 4, then In this paper, we present the local antimagic chromatic number of unicyclic graphs such kite and cycle with two neighbour pendants.A graph is called unicyclic if it is connected and contains exactly one cycle.Therefore, a graph is unicyclic if and only if it is connected and has size equal to its order [4].A kite, denoted by Kt n,m , consists of a cycle of length n with a m-edge path (the tail) attached to one vertex [2].

Main Results
We start this section with a new result on the local antimagic chromatic number of the kite graph in the following theorem.
for even m and even i for even m for odd m and odd j m−j 2 for even m and even j m−1 2 + j 2 + n + 1 for odd m and even j m 2 + j+1 2 + n for even m and odd j It is easy to see that f is a local antimagic labeling and the weight of vertices are for even m and i = 1 n + m + 1 for even (m + i) n + m for odd (m + i)  We note that a n-pan graph, denoted by P g n , is the graph obtained by joining a cycle graph C n to a singleton graph K 1 with a bridge.In other words, the n-pan graph is a special case of the kite graph Kt n,m when m = 1.Consequently, Corollary 2.1.For the n-pan graph P g n with n ≥ 3, χ la (P g n ) = 3.
In the next theorem, we present the local antimagic chromatic number of another unicyclic graph, that is the cycle with two neighbour pendants, as follows.
It is easy to see that f is a local antimagic labeling and the weight of vertices are Thus χ la (Cp n ) ≤ 4. To show the lower bound, we suppose that f (u Figure 2 shows an example of the local antimagic vertex coloring of the cycle with two neighbour pendants Cp 6 with the local antimagic chromatic number equals to 4.

Conclusion
Another family of unicyclic graph is a sun.A sun, denoted by Su n , is a cycle on n vertices C n with an edge terminating in a vertex of degree 1 attached to each vertex [2].The local antimagic chromatic number of the sun Su n has not been discovered.Consequently, we have the following open problems.

Theorem 2 . 1 .
For the kite Kt n,m with n ≥ 3 and m ≥ 1, χ la (Kt n,m ) = 3.Proof.Let Kt n,m be the kite with n ≥ 3 and m

3m+3 2 + n 2 + n for odd m and i = 1 w
(v i ) = n + m for even (m + j) n + m + 1 for odd (m + j) Thus, χ la (Kt n,m ) ≤ 3. To show the lower bound, we can use the local antimagic chromatic number of cycle C n due to Arumugam et al. [1].Since for n ≥ 3, χ la (C n ) = 3 and the kite Kt n,m contains a cycle C n , it easy to see that χ la (Kt n,m ) ≥ 3. Therefore χ la (Kt n,m ) = 3.

Figure 1
Figure 1 shows an example of the local antimagic vertex coloring of the kite Kt 5,6 with the local antimagic chromatic number equals to 3.We note that a n-pan graph, denoted by P g n , is the graph obtained by joining a cycle graph C n to a singleton graph K 1 with a bridge.In other words, the n-pan graph is a special case of the kite graph Kt n,m when m = 1.Consequently,

Problem 1 .
Determine the local antimagic chromatic number of sun.