Abstract:
A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl $k$-L(2,1)-labeling} of a graph $G$ is a mapping $f$ from $V(G)$ to the set ${0,1,...,k}$ such that $|f(u)-f(v)|\ge 2$ if $d_G(u,v)=1$ and $|f(u)-f(v)|\ge 1$ if $d_G(u,v)=2$. The {\sl L(2,1)-labeling number $\lambda (G)$} of $G$ is the smallest number $k$ such that $G$ has a $k$-L(2,1)-labeling. The purpose of this paper is to study path covering number and L(2,1)-labeling number of graphs. Our main work extends most of results in [On island sequences of labelings with a condition at distance two, Discrete Applied Maths 158 (2010), 1-7] and can answer an open problem in [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005), 208-223].

Abstract:
For the graph $G_{1}$ and $G_{2}$ the tensor product is denoted by $G_{1}(T_{p})G_{2}$ which is the graph with vertex set $V(G_{1}(T_{p})G_{2}) = V(G_{1}) imes V(G_{2})$ and edge set $E(G_{1}(T_{p})G_{2})= {(u_{1},v_{1}),(u_{2},v_{2})/u_{1}u_{2} epsilon E(G_{1})$ and $v_{1}v_{2} epsilon E(G_{2})}$. The graph $P_{m}(T_{p})P_{n}$ is disconnected for $forall m,n$ while the graphs $C_{m}(T_{p})C_{n}$ and $C_{m}(T_{p})P_{n}$ are disconnected for both $m$ and $n$ even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling.

Abstract:
In this paper we investigate product cordial labeling for some new graphs. We prove that the friendship graph, cycle with one chord (except when n is even and the chord joining the vertices at diameter distance), cycle with twin chords (except when n is even and one of the chord joining the vertices at diameter distance) are product cordial graphs. We also investigated middle graph of path $P_{n}$ admits product cordial labeling.

Abstract:
L(2,1)-labeling number of the product and the join graph on two fans are discussed in this paper, we proved that L(2,1)-labeling number of the product graph on two fans is λ(G) ≤ Δ+3 , L(2,1)-labeling number of the join graph on two fans is λ(G) ≤ 2Δ+3.

Abstract:
An (2,1) L -labeling of a graph G is a function f from the vertex set V (G) to the set of all nonnegative integers such that |f(x)-f(y)| >= 2 if d(x,y) = 1 and |f(x)-f(y)| >= 1 if if d(x,y) = 2. The L(2,1) -labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v): v∈ V(G). In this paper we completely determine λ-number for middle graph of path Pn, cycle Cn , star K1,n , friendship graph Fn and wheel Wn.

Abstract:
The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $\Delta\geq12$ and $d=2$, the (2,1)-total labelling number $\lambda_2^T(G)$ is at most $\Delta+2$.

Abstract:
We investigate E-cordial labeling for some cartesian product of graphs. We prove that the graphs Kn × P2 and Pn × P2 are E-cordial for n even while Wn × P2 andK1,n × P2 are E-cordial for n odd. Key words: E-Cordial labeling; Edge graceful labeling; Cartesian product

Abstract:
A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident to $x$, and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are a pair of adjacent vertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$. The $(2,1)$-total labeling number $\lambda^T_2(G)$ of a graph $G$ is defined as the minimum $k$ among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585--2593 (2007)], Chen and Wang conjectured that all outerplanar graphs $G$ satisfy $\lambda^T_2(G) \leq \Delta(G)+2$, where $\Delta(G)$ is the maximum degree of $G$, while they also showed that it is true for $G$ with $\Delta(G)\geq 5$. In this paper, we solve their conjecture completely, by proving that $\lambda^T_2(G) \leq \Delta(G)+2$ even in the case of $\Delta(G)\leq 4 $.

Abstract:
For a graph having no isolated vertex, a function is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.

Abstract:
A new concept of labeling called the signed product cordial labeling is introduced and investigated for path graph, cycle graphs, star-K_{1,n}, Bistar-B_{n,n}, and Some general results on signed product cordial labeling are studied.