Given a loopless multigraph $G$ and a vertex-function $f\ : \ V(G) \rightarrow \mathbb{N}\backslash \{0\}$, an {\bf $f$-(edge)-coloring} of $G$ is an assignment of colors to the edges of $G$ such that each color appears at each vertex $v\in V(G)$ at most $f(v)$ times. The {\bf $f$-chromatic index} of a graph $G$, denoted by $\chi'_f(G)$, is the least integer $k$ such that $G$ admits an $f$-coloring using $k$ colors. Clearly, a proper graph edge-coloring is a special $f$-coloring, where $f\equiv 1$. By definition, the chromatic index $\chi'(G)$ is exactly $\chi_1'(G)$.

The {\bf total chromatic number} $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The full text will be divided into the following three chapters.

In the first chapter, we first give some basic definitions, notation, and related terminologies that this dissertation needs. Then we introduce the research background and research significance of this dissertation, as well as the main results of this paper.

In the second Chapter, we mainly study the relationship $\chi''(G)$ and $\chi'(G)$. By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$.

In 1984, Goldberg conjectured that for any multigraph $G$,

if $\chi'(G) \ge \D(G) +3$ then $\chi''(G) = \chi'(G)$.

In this chapter, we show that Goldberg's conjecture is asymptotically true. More specifically, we prove that for a multigraph $G$ with maximum degree $\D$ sufficiently large, if $\chi'(G) \ge \D + 10\D^{35/36}$, then we have $\chi''(G) = \chi'(G)$.

In the third Chapter, we confirm the Goldberg-Seymour conjecture for$f$-coloring, which states that $\chi'_f(G) \le \max\{ \D_f(G) +1, \omega_f(G)\}$, where $f$-maximum degree $\D_f(G)$ and $f$-density $\omega_f(G)$ of a weighted graph $(G, f)$ are defined as $\max_{v\in V(G)} \left\lceil{\frac{d(v)}{f(v)}}\right\rceil$ and

$ \max \left\lceil{ \frac{|E(H)|} {\lfloor \sum_{v\in V(H)}f(v)/2}\rfloor}\right\rceil$, respectively. Our result

implies that $\chi_f'(G)$ can only assume one of two consecutive integers :

$\max\{\D_f(G), \omega_f(G)\}$ and $\max\{\D_f(G) +1, \omega_f(G)\}$.

So an analog to Vizing's theorem on proper edge-colorings of simple

graphs holds for $f$-coloring of all multigraphs.