On the Adjacency Matrix and the Colourings of Graphs
On the Adjacency Matrix and the Colourings of Graphs
Summary: A planar graph, \(G\), can be drawn on a plane in such a way that no two edges intersect. It is said to be maximal planar if no edge can be added without losing planarity. Each vertex of an Eulerian graph is of even degree. We show that the chromatic number of a maximal planar graph is 3 if and only if it is Eulerian. From the adjacency matrix of a planar graph, the monochromatic classes can be deduced and unique colourability determined. Moreover, we show that certain transformations on a graph \(G\) determine the chromatic number.
- University of Messina Italy
Eulerian and Hamiltonian graphs, Coloring of graphs and hypergraphs, adjacency matrix, Graphs and linear algebra (matrices, eigenvalues, etc.), chromatic number, planar graph, colourability, planarity, Eulerian graph
Eulerian and Hamiltonian graphs, Coloring of graphs and hypergraphs, adjacency matrix, Graphs and linear algebra (matrices, eigenvalues, etc.), chromatic number, planar graph, colourability, planarity, Eulerian graph
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