Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.Example − |{1, 4, 3, 5}| = 4, |{1, 2, 3, 4, 5, ....}| = ∞If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ ... Read More

Graph coloring is the procedure of assignment of colors to each vertex of a graph G such that no adjacent vertices get same color. The objective is to minimize the number of colors while coloring a graph. The smallest number of colors required to color a graph G is called its chromatic number of that graph. Graph coloring problem is a NP Complete problem.Method to Color a GraphThe steps required to color a graph G with n number of vertices are as follows −Step 1 − Arrange the vertices of the graph in some order.Step 2 − Choose the first ... Read More

A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. The third and final chapter of this part highlights the important aspects of functions.Function - DefinitionA function or mapping (Defined as f: X → Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.Function ... Read More

Problem StatementFind the number of spanning trees in the following graph.SolutionThe number of spanning trees obtained from the above graph is 3. They are as follows −These three are the spanning trees for the given graphs. Here the graphs I and II are isomorphic to each other. Clearly, the number of non-isomorphic spanning trees is two.

Problem StatementLet 'G' be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.SolutionBy the sum of degrees theorem, 20 ∑ i=1  deg(Vi) = 2|E|20(3) = 2|E||E| = 30By Euler’s formula,|V| + |R| = |E| + 220+ |R| = 30 + 2|R| = 12Hence, the number of regions is 12.

Problem StatementHow many simple non-isomorphic graphs are possible with 3 vertices?SolutionThere are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.

Problem StatementWhat is the matching number for the following graph?SolutionNumber of vertices = 9We can match only 8 vertices.Matching number is 4.

Problem StatementWhat is the line covering number for the following graph?SolutionNumber of vertices = |V| = n = 7Line covering number = (α1) ≥ ⌈ n / 2 ⌉ = 3α1 ≥ 3By using 3 edges, we can cover all the vertices.Hence, the line covering number is 3.

Problem StatementWhat is the chromatic number of complete graph Kn?SolutionIn a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number Kn = n.

A graph is traversable if you can draw a path between all the vertices without retracing the same path. Based on this path, there are some categories like Euler’s path and Euler’s circuit which are described in this chapter.Euler’s PathAn Euler’s path contains each edge of ‘G’ exactly once and each vertex of ‘G’ at least once. A connected graph G is said to be traversable if it contains an Euler’s path.ExampleEuler’s Path = d-c-a-b-d-e.Euler’s CircuitIn an Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.ExampleEuler’s Path = a-b-c-d-a-g-f-e-c-a.Euler’s Circuit ... Read More