Simpler and More General Distributed Coloring Based on Simple List Defective Coloring Algorithms

TL;DR

This paper advances distributed graph coloring by integrating list colorings with defective and arbdefective variants through an oriented list defective coloring (OLDC) framework. It introduces a simple two-sweep OLDC algorithm that, under explicit slack conditions, colors nodes from lists while bounding outneighbor conflicts, achieving efficient runtimes in both LOCAL and CONGEST models. The work extends OLDC to graphs with bounded neighborhood independence $ heta$, via a recursive arbdefective-to-defective coloring strategy and color-space reductions, yielding near-optimal CONGEST round complexities for $( ext{Delta}+1)$-coloring when $ heta=O(1)$, including $ ilde{O}( oot ext{log} extDelta) ext{ bounds}$. The results improve practical distributed coloring performance, enabling faster $( extDelta+1)$-colorings and related colorings in networks with structured neighborhoods, and provide a simpler, more computationally efficient alternative to prior FK23/MausT20-type methods.

Abstract

In this paper, we give list coloring variants of simple iterative defective coloring algorithms. Formally, in a list defective coloring instance, each node $v$ of a graph is given a list $L_v$ of colors and a list of allowed defects $d_v(x)$ for the colors. Each node $v$ needs to be colored with a color $x\in L_v$ such that at most $d_v(x)$ neighbors of $v$ also pick the same color $x$. For a defect parameter $d$, it is known that by making two sweeps in opposite order over the nodes of an edge-oriented graph with maximum outdegree $β$, one can compute a coloring with $O(β^2/d^2)$ colors such that every node has at most $d$ outneighbors of the same color. We generalize this and show that if all nodes have lists of size $p^2$ and $\forall v:\sum_{x\in L_v}(d_v(x)+1)>p\cdotβ$, we can make two sweeps of the nodes such that at the end, each node $v$ has chosen a color $x\in L_v$ for which at most $d_v(x)$ outneighbors of $v$ are colored with color $x$. Our algorithm is simpler and computationally significantly more efficient than existing algorithms for similar list defective coloring problems. We show that the above result can in particular be used to obtain an alternative $\tilde{O}(\sqrtΔ)+O(\log^* n)$-round algorithm for the $(Δ+1)$-coloring problem in the CONGEST model. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. It is known that by doing a single sweep over the nodes of a graph of neighborhood independence $θ$, one can compute a $d$-defective coloring with $O(θ\cdot Δ/d)$ colors. We extend this approach to the list defective coloring setting and use it to obtain an efficient recursive coloring algorithm for graphs of neighborhood independence $θ$. In particular, if $θ=O(1)$, we get an $(\logΔ)^{O(\log\logΔ)}+O(\log^* n)$-round algorithm.

Figures (1)

• Figure 1: The picture illustrates a node $v$ (in red) together with its outneighbors in $N_<(v)$ (in blue) and $N_>(v)$ (in green). Note that when node $v$ has to pick the set $S_v$ in Phase $I$, all blue nodes $u$ have already picked their subset $S_u$ of their color list and have sent it to $v$. In Phase $II$, $v$ has to output a final color from $S_v$. At this point, the green nodes ($N_<(v)$) have already picked their final colors. Node $v$ therefore has to pick its color based on knowing the setst $S_u$ for nodes in $N_<(v)$ and the final colors of nodes in $N_>(v)$.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.1: Slack
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof