there exists a sequence-number function $s_f$ for $f$;
$g$ is moderately-fast;
the dependence of $f$ on $m$ is bounded by a polylog; and
the dependence of $f$ on $\Delta$ is moderately-slow;
then there exists a uniform $O(g(\Delta))$-coloring algorithm running in time $O(f(\Lambda^*) \cdot s_f(f(\Lambda^*)))$.
Proof Our first goal is to obtain a coloring algorithm that does not require $m$ (and thus requires only $\Delta$). For this purpose we first define the following problem. The strong list-coloring (SLC) problem: a configuration for the SLC problem is a pair $(G, \mathbf{x}) \in \mathcal{F}(\mathcal{G})$ such that
(1) there exists an integer $\hat{\Delta}$ in $\cap_{v \in V(G)} \mathbf{x}(v)$ such that $\hat{\Delta} \ge \Delta$; and
(2) the input $\mathbf{x}(v)$ of every vertex $v \in V(G)$ contains a list $L(v)$ of colors contained in $[1, g(\hat{\Delta})) \times [1, \hat{\Delta}+1]$ such that
Given a configuration $(G, \mathbf{x}) \in \mathcal{F}(\mathcal{G})$, an output vector $\mathbf{y}$ is a solution to SLC if it forms a coloring and if $\mathbf{y}(v) \in L(v)$ for every node $v \in V(G)$. Condition (1) above implies that a local algorithm for SLC can use an upper bound on $\Delta$, which is the same for all nodes. Informally, Condition (2) above implies that the list $L(v)$ of colors available for each node $v$ contains $\deg_G(v)+1$ copies of each color in the range $[1, g(\hat{\Delta}))$.
We now design a pruning algorithm $\mathcal{P}$ for SLC. Consider a triplet $(G, \mathbf{x}, \hat{\mathbf{y}})$, where $(G, \mathbf{x})$ is a configuration for SLC and $\hat{\mathbf{y}}$ is some tentative assignment of colors. The set $W$ of nodes to be pruned is composed of all nodes $u$ satisfying $\hat{\mathbf{y}}(u) \in L(u)$ and $\hat{\mathbf{y}}(u) \neq \hat{\mathbf{y}}(v)$ for all $v \in N_G(u)$. For each node $u \in V \setminus W$, set
In other words, $L'(u)$ contains all the colors in $L(u)$ that are not assigned to a neighbor of $u$ belonging to $W$. Algorithm $\mathcal{P}$ returns the configuration $(G', \mathbf{x}')$, where $G'$ is the subgraph of $G$ obtained by removing the nodes in $W$ and
Observe that if we start with a configuration $(G, \mathbf{x})$ for SLC, then the output $(G', \mathbf{x}')$ of the pruning algorithm $\mathcal{P}$ is also a configuration for SLC. Indeed, for every node $v$ and every integer $k$, at most $\deg_W(v)$ pairs $(k, j)$ are removed from the list $L(v)$ of $v$, where $\deg_W(v)$ is the number of neighbors of $v$ that belong to $W$. On the other hand, the degree of $v$ in $G'$ is reduced by $\deg_W(v)$. Note also that the input vector of all nodes
still contain $\hat{\Delta}$, which is an upper bound for the maximum degree of $G'$.
Starting with $\mathcal{A}^\Gamma$, it is straightforward to design a local algorithm $\mathcal{B}^{\Gamma'}$ for SLC that depends on $\Gamma' = \Gamma \setminus {\Delta}$. Specifically, $\mathcal{B}^{\Gamma'}$ executes $\mathcal{A}^\Gamma$ using the good guess $\hat{\Delta} = \hat{\Delta}$ for the parameter $\Delta$. Furthermore, if $\mathcal{A}^\Gamma$ outputs at $v$ a color $c$, then $\mathcal{B}^{\Gamma'}$ outputs the color $(c, j)$ where $j = \min{s : (c, s) \in L(v)}$.
Given an instance for SLC, we view $\hat{\Delta}$ as a non-decreasing parameter, and convert $\Lambda$ to a new set of non-decreasing parameters $\Lambda'$ by replacing $\Delta$ with $\Delta'$. Formally, if $\Delta \in \Lambda$ then set $\Lambda' = (\Lambda \setminus \Delta) \cup \hat{\Delta}$, and otherwise, set $\Lambda' = \Lambda$. Since $\Gamma$ and $\Lambda$ contain only non-decreasing graph-parameters—and since $\hat{\Delta}$ is contained in all the inputs—we deduce that the pruning algorithm $\mathcal{P}$ is $(\Gamma' \cup \Lambda')$-monotone.
Now, we apply Theorem 3 to Algorithm $\mathcal{B}^{\Gamma'}$, the sets $\Gamma'$ and $\Lambda'$ of non-decreasing parameters and the aforementioned pruning algorithm $\mathcal{P}$ for SLC. We obtain a uniform algorithm $\mathcal{B}$ for SLC and $\mathcal{F}(\mathcal{G})$, whose running time is $O(f(\Lambda'^*) \cdot s_f(f(\Lambda'^*)))$.
We are ready to specify the desired uniform $O(g(\Delta))$-coloring algorithm. We define inductively a sequence $(D_i)_{i \in \mathbb{N}}$ by setting $D_1 = 1$ and
for $i \ge 1$. As $g$ is moderately-increasing, there is a constant $\alpha$ such that for each integer $i \ge 1$,
$D_{i+1} \ge \alpha D_i$ and
$g(D_{i+1}) \le \alpha^{\log \alpha} g(D_i)$.
Given an initial configuration $(G, \mathbf{x})$, we partition it according to the node degrees. For $i \in \mathbb{N}$, let $D_i$ be the subgraph of $G$ induced by the set of nodes $v$ of $G$ with $\deg_G(v) \in [D_i, D_{i+1}-1]$. Let $\mathbf{x}_i$ be the input $\mathbf{x}$ restricted to the nodes in $D_i$. The configuration $(G_i, \mathbf{x}i)$, which belongs to $\mathcal{F}(\mathcal{G})$, is referred to as layer i. Note that nodes can figure out locally which layer they belong to. Observe also that $D{i+1}-1$ is an upper bound on node degrees in layer i.
The algorithm proceeds in two phases. In the first phase, each node in layer i is assigned the list of colors $L_i'' = [1, g(D_{i+1})] \times [1, D_{i+1}+1]$, and the degree estimation $\hat{\Delta}i = D{i+1}$. Each layer is now an instance of SLC and we execute Algorithm B in parallel on all layers. If Algorithm B assigns a color $(c, j)$ to a node $v$ in layer i then we change this color to $(g(D_{i+1}) + c, j)$. Hence, for each i, layer i is colored with colors taken from $L_i' = [g(D_{i+1}) + 1, 2g(D_{i+1})] \times [1, D_{i+1}+1]$.
Note that nodes in different layers have disjoint color lists, and hence we obtain a coloring of the whole graph G. The number of colors in $L_i'$ is at most $2D_{i+1}g(D_{i+1})$.