An equitable coloring of a graph is a proper coloring where the sizes of any two different color classes do not differ by more than one. We use $\mathcal{G}_{m_1, m_2}$ to represent the class of graphs $G$ that satisfy the following conditions: for any subgraph $H$ of $G$, the inequality $e(H) \leq m_1 v(H)$ holds, and for any bipartite subgraph $H$ of $G$, the inequality $e(H) \leq m_2 v(H)$ holds. A graph $G$ is $\alpha$-sparse if $e(H) \leq \alpha v(H)$ for every subgraph $H$ of $G$. In this paper, we show that there is a small constant $r_0\in [4m_1, 6.21m_1]$ solely determined by both $m_1$ and $m_2$, such that for any graph $G\in \mathcal{G}_{m_1, m_2}$ (where the ratio $m_1/m_2$ is between $1$ and $1.8$ inclusive) with a maximum degree $\Delta(G)\geq r_0$, an equitable $r$-coloring is guaranteed for all $r\geq \Delta(G)$. By setting $m_1=m_2=\alpha$ in this result, we conclude that every $\alpha$-sparse graph $G$ has an equitable $r$-coloring for every $r\geq \Delta(G)$ provided $\Delta(G)\geq 6.21\alpha$. Consequently, the celebrated Equitable $\Delta$-Color Conjecture and Chen-Lih-Wu Conjecture are verified for sparse graphs with large maximum degree. The local crossing number of a drawing of a graph is the largest number of crossings on a single edge, and the local crossing number of that graph is the minimum of such values among all possible drawings. As an interesting application of our main result, we confirm Equitable $\Delta$-Color Conjecture and Chen-Lih-Wu Conjecture for non-planar graphs $G$ with local crossing number not exceeding $\Delta(G)^2 / 383$.

翻译：图的公平着色是一种正常着色，其中任意两个不同颜色类的大小相差不超过一。我们用 $\mathcal{G}_{m_1, m_2}$ 表示满足以下条件的图 $G$ 的类：对于 $G$ 的任何子图 $H$，不等式 $e(H) \leq m_1 v(H)$ 成立；对于 $G$ 的任何二分子图 $H$，不等式 $e(H) \leq m_2 v(H)$ 成立。如果对于图 $G$ 的每个子图 $H$ 都有 $e(H) \leq \alpha v(H)$，则称图 $G$ 是 $\alpha$-稀疏的。在本文中，我们证明存在一个仅由 $m_1$ 和 $m_2$ 决定的小常数 $r_0\in [4m_1, 6.21m_1]$，使得对于任何属于 $\mathcal{G}_{m_1, m_2}$ 的图 $G$（其中比率 $m_1/m_2$ 在 $1$ 到 $1.8$ 之间，包含端点）且最大度 $\Delta(G)\geq r_0$，保证对所有 $r\geq \Delta(G)$ 存在公平的 $r$-着色。在此结果中令 $m_1=m_2=\alpha$，我们得出结论：每个 $\alpha$-稀疏图 $G$ 在 $\Delta(G)\geq 6.21\alpha$ 的条件下，对每个 $r\geq \Delta(G)$ 都存在公平的 $r$-着色。因此，著名的公平 $\Delta$-着色猜想和 Chen-Lih-Wu 猜想对于具有大最大度的稀疏图得到了验证。图的绘制的局部交叉数是指单条边上的最大交叉数，而该图的局部交叉数是在所有可能绘制中该值的最小值。作为我们主要结果的一个有趣应用，我们证实了对于局部交叉数不超过 $\Delta(G)^2 / 383$ 的非平面图 $G$，公平 $\Delta$-着色猜想和 Chen-Lih-Wu 猜想成立。