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# What is the chromatic number of a Petersen graph?

## What is the chromatic number of a Petersen graph?

3

Petersen graph
Chromatic number 3
Chromatic index 4
Fractional chromatic index 3
Genus 1

## What is chromatic polynomial in graph theory?

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem.

Is Petersen graph Euler?

1 The Petersen graph is non-hamiltonian. It follows that every Hamilton cycle of G must contain four edges in T. A digraph G is Eulerian ⇔ L(G) is hamiltonian. ⇐ does not hold for undirected graphs, for example, a star K1,3.

Is the Petersen graph 3 colorable?

The Petersen graph is not 3-edge-colorable. The Petersen graph is usually drawn as an outer 5-cycle, an inner 5-cycle where edges join vertices that are cyclically two apart, and a matching joining corresponding vertices on the two cycles drawn as depicted in Fig. 1.

### What is chromatic index of a graph?

The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.

### How many cycles does Petersen graph have?

Petersen Graph

property value
Hamiltonian graph no
Hamiltonian cycle count 0
Hamiltonian path count 240
hypohamiltonian graph yes

What is chromatic 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.

What is Euler graph with example?

Thus, start at one even vertex, travel over each vertex once and only once, and end at the starting point. One example of an Euler circuit for this graph is A, E, A, B, C, B, E, C, D, E, F, D, F, A. This is a circuit that travels over every edge once and only once and starts and ends in the same place.

## What is Dodecahedral graph?

The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, illustrated above in four embeddings. It is also a unit-distance graph (Gerbracht 2008), as shown above in a unit-distance embedding.

## How do you find the chromatic index of a graph?

The chromatic index χ ΄ (x) is the minimum number of different colors needed to color edges such that any two adjacent edges are colored by different colors (for more details, see [1, 3,4,5, 7–9, 11,12,13,14]). Kӧnig has proved, in 1916, that χ ΄ (x) = ∆(x) for every bipartite graph.

How do you find the chromatic index?

The chromatic index of a graph G, denoted x'(G), is the minimum number of colors used among all colorings of G. Vizing [l l] has shown that for any graph G, x'(G) is either its maximum degree A(G) or A(G) + 1. If x'(G) = A(G) then G is in Class 1; otherwise G is in Class 2.