algorithm requires that, in particular, the nodes can locally decide whether they belong to a legal configuration. While locally checking that neighboring nodes have distinct colors is easy, knowing whether a color is in the required range, namely, $[1, O(\Delta)]$, seems difficult as the nodes do not know $\Delta$. Moreover, the gluing property seems difficult to tackle also: after pruning a node with color $c$, none of its unpruned neighbors can be colored in color $c$. In other words, a correct solution on the non-pruned subgraph may not glue well with the pruned subgraph.
Nevertheless, we show in this section that several relatively general transformers can be used to obtain uniform coloring algorithms from non-uniform one. We focus on standard coloring problems in which the required number of colors is given as a function of $\Delta$.
5.1 Uniform ($\Delta + 1$)-coloring Algorithms
A standard trick (cf., [28,30]) allows us to transform an efficient (with respect to $n$ and $\Delta$) MIS algorithm for general graphs into one for ($\Delta + 1$)-coloring (and, actually, to the more general maximal coloring problem defined by Luby [30]). The general idea is based on the observation that ($\Delta + 1$)-colorings of $G$ and maximal independent sets of $G' = G \times K_{\Delta+1}$ are in one-to-one correspondence. More precisely, and avoiding the use of $\Delta$, the graph $G'$ is constructed from $G$ as follows. For each node $u \in V(G)$, take a clique $C_u$ of size $\deg_G(u)+1$ with nodes $u_1, \dots, u_{\deg_G(u)+1}$. Now, for each $(u,v) \in E(G)$ and each $i \in [1, 1+\min{\deg_G(u), \deg_G(v)})$, let $(u_i, v_i) \in E(G')$. The graph $G'$ can be constructed by a local algorithm without using any global parameter. It remains to observe the existence of a natural one-to-one correspondence between the maximal independent sets of $G'$ and the ($\deg_G+1$)-colorings of $G$, that is, the colorings of $G$ such that each node $u$ is assigned a color in $[1, \deg_G(u)+1]$.
To see this, first consider a ($\deg_G+1$)-coloring $c$ of $G$. Set
Then, no two nodes in $X$ are adjacent in $G'$. Moreover, a node that does not belong to $X$ has a neighbor in $X$ since $X$ contains a vertex from each clique $C_u$ for $u \in V(G)$. Therefore, $X$ is a MIS of $G'$.
Conversely, let $X$ be a MIS of $G'$. We assert that $X$ contains a node from every clique $C_u$ for $u \in V(G)$. Indeed, suppose on the contrary that $X \cap V(C_u) = \emptyset$ for a node $u \in V(G)$. By the definition of a MIS, every vertex $u_i \in V(C_u)$ has a neighbor $v(u_i)$ that belongs to $X$. Since a clique can contain at most one node in $X$
and $v(u_i) \neq v(u_j)$ whenever $i \neq j$, we deduce that at least $|C_u|$ cliques $C_v$ with $v \neq u$ contain a node that has a neighbor in $C_u$. This contradicts the definition of $G'$, since $|C_u| = \deg_G(u) + 1$. Thus, setting $c(u)$ to be the index $i \in {1, \dots, \deg_G(u) + 1}$ such that $u_i \in X$ yields a ($\deg_G+1$)-coloring of $G$.
Therefore, we obtain Corollary 1(ii) as a direct consequence of Corollary 1(i).
5.2 Uniform Coloring with More than $\Delta + 1$ Colors
We now aim to provide a transformer taking as input an efficient non-uniform coloring algorithm that uses $g(\Delta)$ colors (where $g(\Delta) > \Delta$) and produces an efficient uniform coloring algorithm that uses $O(g(\Delta))$ colors. We begin with the following definitions.
An instance for the coloring problem is a pair $(G, \mathbf{x})$ where $G$ is a graph and $\mathbf{x}(v)$ contains a color $c(v)$ such that the collection ${c(v) : v \in V(G)}$ forms a coloring of $G$. (The color $c(v)$ can be the identity $\mathrm{Id}(v)$.) For a given family $\mathcal{G}$ of graphs, we define $\mathcal{F}(\mathcal{G})$ to be the collection of instances $(G, \mathbf{x})$ for the coloring problem, where $G \in \mathcal{G}$.
Many coloring algorithms consider the identities as colors, and relax the assumption that the identities are unique by replacing it with the weaker requirement that the set of initial colors forms a coloring. Given an instance $(G, \mathbf{x})$, let $m = m(G, \mathbf{x})$ be the maximal identity. Note that $m$ is a graph-parameter.
Recall the $\lambda(\tilde{\Delta}+1)$-coloring algorithms designed by Barenboim and Elkin [4] and Kuhn [22] (which generalize the $\mathcal{O}(\tilde{\Delta}^2)$-coloring algorithm of Linial [28]). We would like to point out that, in fact, everything works similarly in these algorithms if one replaces $n$ with $m$. That is, these $\lambda(\tilde{\Delta}+1)$-coloring algorithms can be viewed as requiring $m$ and $\Delta$ and running in time $\mathcal{O}(\tilde{\Delta}/\lambda + \log^*\tilde{m})$. The same is true for the edge-coloring algorithms of Barenboim and Elkin [7].
The following theorem implies that these algorithms can be transformed into uniform ones. In the theorem, we consider two sets $\Gamma$ and $\Lambda$ of non-decreasing graph-parameters such that
(1) $\Gamma$ is weakly-dominated by $\Lambda$; and
(2) $\Gamma \subseteq {\Delta, m}$.
Two such sets of parameters are said to be related. The notion of moderately-fast function (defined in Section 2) will be used to govern the number of colors used by the coloring algorithms.
Theorem 5 Let $\Gamma$ and $\Lambda$ be two related sets of non-decreasing graph-parameters and let $\mathcal{A}^\Gamma$ be a $\mathcal{g}(\tilde{\Delta})$-coloring algorithm with running time bounded with respect to $\Lambda$ by some function $f$. If