The paper by W. T. Tutte introduces and studies two polynomials, $\theta(G, n)$ and $\phi(G, n)$, related to graph colorings and associated maps. $\theta(G, n)$ and $\phi(G, n)$ are defined in terms of the number of $n$-colorings of a graph $G$. The author proves a new result for $\phi(G, n)$ and discusses unsolved problems related to it, which are natural generalizations of the Four Color Problem from planar graphs to general graphs. Additionally, a polynomial $\chi(G, x, y)$ is introduced, which generalizes both $\theta(G, n)$ and $\phi(G, n)$. $\chi(G, x, y)$ is defined in terms of spanning trees of $G$ and is shown to be invariant under changes in the enumeration of the edges. The paper also explores the connection between the theory of spanning trees and electrical networks, highlighting the importance of $\chi(G, x, y)$ as a generalization of the number of spanning trees $C(G)$. The results provide a framework for understanding and computing chromatic polynomials and their properties.The paper by W. T. Tutte introduces and studies two polynomials, $\theta(G, n)$ and $\phi(G, n)$, related to graph colorings and associated maps. $\theta(G, n)$ and $\phi(G, n)$ are defined in terms of the number of $n$-colorings of a graph $G$. The author proves a new result for $\phi(G, n)$ and discusses unsolved problems related to it, which are natural generalizations of the Four Color Problem from planar graphs to general graphs. Additionally, a polynomial $\chi(G, x, y)$ is introduced, which generalizes both $\theta(G, n)$ and $\phi(G, n)$. $\chi(G, x, y)$ is defined in terms of spanning trees of $G$ and is shown to be invariant under changes in the enumeration of the edges. The paper also explores the connection between the theory of spanning trees and electrical networks, highlighting the importance of $\chi(G, x, y)$ as a generalization of the number of spanning trees $C(G)$. The results provide a framework for understanding and computing chromatic polynomials and their properties.