It is proved that if the number of colors $c$ is not a perfect square, the state-of-the-art defect for distributed $c-colorings by a constant factor in most cases can be improved.
Abstract
A $d$-defective $c$-coloring of a graph $G=(V,E)$ is a coloring of the nodes $V$ with $c$ colors such that every node has at most $d$ neighbors of the same color. Distributed algorithms for computing different variants of defective coloring are at the core of most deterministic state-of-the-art distributed coloring algorithms, and they are also an important tool in many other distributed graph algorithms. In several cases, the overall complexity could be improved if some version of defective coloring could be solved more efficiently. Barenboim and Elkin [STOC'09] introduced a two-pass greedy algorithm that uses $p^2$ colors with defect $\lfloor \Delta/p\rfloor$ in $O(\Delta+\log^{\ast} n)$ rounds. This remains the best defect/color tradeoff for $O(\log^{\ast} n)$-time algorithms in bounded-degree graphs. This paper expands the capabilities of this two-pass algorithm. First, we generalize it to the \emph{list defective coloring} problem (Fuchs and Kuhn, [DISC'23]). Consequently, we obtain an alternative algorithm for computing a proper $(\Delta+1)$-coloring in $\tilde{O}(\sqrt{\Delta}) + O(\log^{\ast} n)$ rounds in the CONGEST model. Second, we analyze a generalized two-pass algorithm for standard defective colorings. We prove that if the number of colors $c$ is not a perfect square, we can improve the state-of-the-art defect for distributed $c$-colorings by a constant factor in most cases. However, we also prove a limitation: for any $c\geq 1$, this generalized algorithm cannot achieve a $c$-coloring with defect below $(1-o(1))\cdot\Delta/\sqrt{c}$.
The first $poly(\Delta,\log n)-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model is presented and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.
Sebastian Brandt, Ananth Narayanan, Alexandre Nolin· 0 citations
A suite of fast randomized distributed algorithms representing varying points on this tradeoff are presented, analyze their properties, and study their time complexity in the sequential, CONGEST and Congested Clique models.
This work generalizes and combines tools from the $(k+2)-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023] and yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.
A $(k,d)$-edge coloring of a loopless multigraph $G$ is an edge coloring using at most $k$ colors such that the subgraph formed by each color class has maximum degree at most $d$. The least such $k$ is denoted by $\chi'_d(G)$. Let $G$ be a loopless non-bipartite multigraph with maximum degree $\Delta(G)$ and odd girth $g_0(G)$, and let $d\ge1$ be odd. We prove that \[ \chi'_d(G)\le\left\lceil\frac{g_0(G)\Delta(G)-1}{dg_0(G)-1}\right\rceil. \] For $d=1$, this is Goldberg's odd-girth refinement of Shannon's theorem, while for $g_0(G)=3$ it is the defective Shannon bound of Aboulker, Aubian, and Huang. For every odd $d>1$, every odd $g_0\ge3$, and every $\Delta>d$, an almost full ring multigraph $R(\Delta,g_0)$, an odd cycle with edge multiplicities alternating between $\lfloor\Delta/2\rfloor$ and $\lceil\Delta/2\rceil$, except that two consecutive edges have multiplicity $\lfloor\Delta/2\rfloor$, attains equality. We also derive a range in which the defective Goldberg--Seymour conjecture holds.
A proper conflict-free (PCF) $k$-coloring of a graph $G$ is a proper $k$-coloring such that there exists a color that appears exactly once in the neighborhood of every non-isolated vertex $v\in V(G)$. The PCF chromatic number, denoted by $\chi_{pcf}(G)$, is the least integer $k$ such that there exists a PCF $k$-coloring of $G$. Given a graph $G$ and a positive integer $k$, PCF $k$-COLORABILITY is to decide whether $G$ admits a PCF $k$-coloring. Ahn et al. [Discrete Appl. Math. 377 (2025) 10-17] proved that PCF $k$-COLORABILITY is NP-complete for bipartite graphs. We strengthen this result by proving that PCF $k$-COLORABILITY is NP-complete for perfect elimination bipartite graphs, which is a proper subclass of bipartite graphs. We also show that the PCF chromatic number of a graph cannot be approximated within $O(n^{1-\varepsilon})$ unless P=NP, for any $\varepsilon>0$. On the positive side, we provide linear-time algorithms for PCF $k$-COLORABILITY in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs. We show that $\chi_{pcf}(G)\leq \omega(G)+1$ for block graphs, proper interval graphs, and pseudo-split graphs (except $C_5$), and we characterize all graphs for which the equality holds.
The main result shows that the exact average-case complexity of this fundamental problem is $\Theta(nk)$ for every $k \leq n^{c'}$ and some $c'\in (0, 1)$, and reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input.
Cassandra Marcussen, Edward Pyne, R. Rubinfeld et al.· 0 citations