 # Question: What Makes A Perfect Graph?

## How do you know if a graph is planarity?

Given a graph G = (V,E), G is planar if it admits a drawing such that no two distinct drawn edges intersect except at end points.

Given a graph G = (V,E), is G planar, i.e., can G be drawn in the plane without edge crossings?.

## What is the vertex coloring of a graph?

A vertex coloring is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. The most common type of vertex coloring seeks to minimize the number of colors for a given graph.

## What is the chromatic number of a graph?

The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of. possible to obtain a k-coloring. Minimal colorings and chromatic numbers for a sample of graphs are illustrated above …

## What is k3 graph?

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).

## How many cycles are in a complete graph?

Actually it can have even more – in a complete graph, consider any permutation and its a cycle hence atleast n! cycles. Actually a complete graph has exactly (n+1)! cycles which is O(nn).

## Is k3 bipartite?

Kuratowski’s theorem tells us that, if we can find a subgraph in any graph that is homeomorphic to K 5 K_5 K5 or K 3 , 3 K_{3,3} K3,3, then the graph is not planar, meaning it’s not possible for the edges to be redrawn such that they are none overlapping. …

## Is k3 3 a eulerian?

The graph K3,3 is non-planar. Proof: in K3,3 we have v = 6 and e = 9. If K3,3 were planar, from Euler’s formula we would have f = 5.

## What is the three color problem?

The Three Color Problem is: Under what conditions can the regions of a planar map be colored in three colors so that no two regions with a common boundary have the same color? This paper describes the origin of the Three Color Problem and virtually all the major results and conjectures extant in the literature.

## What is a unique graph?

In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k−1 independent sets.

## What is the clique number of a graph?

The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.

## What does it mean for a graph to be complete?

A complete graph is a graph in which each pair of graph vertices is connected by an edge. The complete graph with graph vertices is denoted and has (the triangular numbers) undirected edges, where. is a binomial coefficient. In older literature, complete graphs are sometimes called universal graphs.

## What is an undirected graph?

An undirected graph is graph, i.e., a set of objects (called vertices or nodes) that are connected together, where all the edges are bidirectional. An undirected graph is sometimes called an undirected network. In contrast, a graph where the edges point in a direction is called a directed graph.

## What makes a graph eulerian?

Definition: A graph is considered Eulerian if the graph is both connected and has a closed trail (a walk with no repeated edges) containing all edges of the graph. Definition: An Eulerian Trail is a closed walk with no repeated edges but contains all edges of a graph and return to the start vertex.

## What is a k4 graph?

Example 19.1: The complete graph K4 consisting of 4 vertices and with an edge between every pair of vertices is planar. Figure 19.1a shows a representation of K4 in a plane that does not prove K4 is planar, and 19.1b shows that K4 is planar. The graphs K5 and K3,3 are nonplanar graphs.

## What is Graph explain?

Graph is a non-linear data structure that is same as the mathematical (discrete mathematics) concept of graphs. It is a collection of nodes (also called as vertices) and edges that connect these vertices. Graphs are used to represent arbitrary relationship among objects. A graph can be directed or undirected.