A graph is said to be cordial if it has a 0-1 labeling that satisfies certain properties. In this paper we show the Cartesian product of a path and a cycle or vice versa are always cordial under some conditions. Also, we prove that the Cartesian product of two paths is cordial.
A radio mean square labeling of a connected graph is motivated by the channel assignment problem for radio transmitters to avoid interference of signals sent by transmitters. It is an injective map h from the set of vertices of the graph G to the set of positive integers N , such that for any two distinct vertices x , y , the inequality d x , y + h x 2 + h y 2 / 2 ≥ dim G + 1 holds. For a particular radio mean square labeling h , the maximum number of h v taken over all vertices of G is called its spam, denoted by rmsn h , and the minimum value of rmsn h taking over all radio mean square labeling h of G is called the radio mean square number of G , denoted by rmsn G . In this study, we investigate the radio mean square numbers rmsn P n and rmsn C n for path and cycle, respectively. Then, we present an approximate algorithm to determine rmsn G for graph G . Finally, a new mathematical model to find the upper bound of rmsn G for graph G is introduced. A comparison between the proposed approximate algorithm and the proposed mathematical model is given. We also show that the computational results and their analysis prove that the proposed approximate algorithm overcomes the integer linear programming model (ILPM) according to the radio mean square number. On the other hand, the proposed ILPM outperforms the proposed approximate algorithm according to the running time.
A simple graph is said to be signed product cordial if it admits ±1 labeling that satisfies certain conditions. Our aim in this paper is to contribute some new results on signed product cordial labeling and present necessary and sufficient conditions for signed product cordial of the sum and union of two fourth power of paths. We also study the signed product cordiality of the sum and union of fourth power cycles The residue classes modulo 4 are accustomed to find suitable labelings for each class to achieve our task. We have shown that the union and the join of any two fourth power of paths are always signed product cordial. Howover, the join and union of fourth power of cycles are only signed codial with some expectional situations.
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