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The function f is called a square graceful labeling of G. Two new definitions namely, odd square graceful labeling and even square graceful labeling of a graph are defined with example. Terms not defined here are used in the sense of Harary[2]. For number theoretic terminology [1] is followed A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. There are several types of graph labeling and a detailed survey is found in [4]. Several authors worked on graceful labeling, odd graceful labeling, even graceful labeling, super graceful labeling and skolem —graceful labeling.

## On Harmonious Coloring of Middle Graph

In graphical terms, the object of the game is to find a spanning cycle in the graph of the dodecahedron, shown in Fig. This remarkable problem can be explained in five minutes by any mathematician to the socalled man in the street.

At the end of the explanation, both will understand the problem, but neither will be able to solve it. The following quotation from the definitive historical article by May [M5] states the Four Color Conjecture and describes its role: [The conjecture states that] any map on a plane or the surface of a sphere can he colored with only four colors so that no two adjacent countries have the same color. Each country must consist of a single connected region, and adjacent countries are those having a boundary line not merely a single point in common.

The consensus is that the conjecture is correct but unlikely to be proved in general. It seems destined to retain for some time the distinction of being both the simplest and most fascinating unsolved problem of mathematics. The Four Color Conjecture has an interesting history, but its origin remains somewhat vague.

There have been reports that Mbius was familiar with this problem in , but it is only definite that the problem was communicated to De Morgan by Guthrie about The first of many erroneous "proofs" of the conjecture was given in by Kempe [K6]. An error was found in by Heawood [H38] who showed, however, that the conjecture becomes true when "four" is replaced by "five.

## IJPAM: Volume 112, No. 5 (2017)

The Four Color Conjecture is a problem in graph theory because every map yields a graph in which the countries including ihe exterior region are the points, and two points are joined by a line whenever the corresponding countries are adjacent. Such a graph obviously can be drawn in the plane without intersecting lines.

Thus, if it is possible to color the points of every planar graph with four or fewer colors so that adjacent points have different colors, then the Four Color Conjecture will have been proved. A map and its corresponding graph. It was pointed out that Lewin was actually dealing with graphs, as indicated by Fig.

## Contents IJPAM: Volume , No. 4 ()

This viewpoint led the psychologists at the Research Center for Group Dynamics to another psychological interpretation of a graph, in which people are represented by points and interpersonal relations by lines. Such relations include love, hate, communication, and power. For number theoretic terminology [1] is followed A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions.

There are several types of graph labeling and a detailed survey is found in [4].

Several authors worked on graceful labeling, odd graceful labeling, even graceful labeling, super graceful labeling and skolem —graceful labeling.

Recently the concept of square graceful labeling was introduced by T. Tharmaraj and P.

Sarasija inthe year As you would expect from any number-crunching tool, the program can perform basic arithmetic operations. If that sounds too easy, don't worry - you'll find yourself in situations that require clever manipulation of the game board.

For experienced users, this would not be confusing, but the less sophisticated could use additional help. This program seems to be targeted at windows users who are just getting started with keyboard shortcuts.

## IJPAM: Volume 120, No. 4 (2018)

Users first choose project servers. Impatient downloaders will certainly appreciate the progression box that appears after clicking a video site or url, letting them track the download percentage.

Clicking on the "create slideshow settings" button will launch the app's setup wizard that will walk you through the process of creating your own slideshow settings, allowing you to choose the size of the image, background, and number of images per screen, among many other options. While these might be understood by advanced users, the average mac user would probably find them difficult to interpret. Then, anytime you want to post something to one or more of these sites, simply click the icon.

Charts When we first ran the program, we selected the evaluate button to start our day trial.