Do just number 6 Determine Whether P4 is an induced subgraph

Do just number 6.

Determine Whether P_4 is an induced subgraph of K_4, 4 If yes then exhibit it If no, then explain why not List all of the unlabeled connected subgraph s of C_34. The concept of complete bipartite graphs can be generalized to define the complete multipartite graph K_r_1, r_2, r_k. This graph consists of k sets of vertices A_1, A_2, A_k, with |A_i| = r_i for each i, where all possible \"interset edges\" are present and no \"intraset edges\" are present. find expressions for the order and size of K_r_1, r_2, r_k.

Solution

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V₁ and V₂ such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph (V₁, V₂, E) such that for every two vertices v₁ V₁ and v₂ V₂, v₁v₂ is an edge in E. A complete bipartite graph with partitions of size |V₁|=m and |V₂|=n, is denoted K_m,n;

The maximum number of edges in an n-vertex triangle free graph is {\\displaystyle \\lfloor n^{2}/4\ floor .}

In other words, one must delete nearly half of the edges in K_n to obtain a triangle-free graph.