We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates.
We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}.
We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for all $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times.
A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree.
Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].
Vizing showed that it suffices to color the edges of a simple graph using $Δ+ 1$ colors, where $Δ$ is the maximum degree of the graph. However, up to this date, no efficient distributed edge-coloring algorithms are known for obtaining such a coloring, even for constant degree graphs. The current algorithms that get closest to this number of colors are the randomized $(Δ+ \tildeΘ(\sqrtΔ))$-edge-coloring algorithm that runs in $\text{polylog}(n)$ rounds by Chang et al. (SODA '18) and the deterministic $(Δ+ \text{polylog}(n))$-edge-coloring algorithm that runs in $\text{poly}(Δ, \log n)$ rounds b…
Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online $(1+o(1))Δ$-edge-coloring algorithm exists for $n$-node graphs of maximum degree $Δ=ω(\log n)$. This conjecture remains open in general, though it was recently proven for bipartite graphs under \emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the \emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound […
We develop sequential algorithms for constructing edge-colorings of graphs and multigraphs efficiently and using few colors. Our primary focus is edge-coloring arbitrary simple graphs using $d+1$ colors, where $d$ is the largest vertex degree in the graph. Vizing's Theorem states that every simple graph can be edge-colored using $d+1$ colors. Although some graphs can be edge-colored using only $d$ colors, it is NP-hard to recognize graphs of this type [Holyer, 1981]. So using $d+1$ colors is a natural goal. Efficient techniques for $(d+1)$-edge-coloring were developed by Gabow, Nishizeki, Kari…
Vizing's celebrated theorem asserts that any graph of maximum degree $Δ$ admits an edge coloring using at most $Δ+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Δ-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Δ=O(\log n)$, and they conjectured the existence of online algorithms using $Δ(1+o(1))$ colors for $Δ=ω(\log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 a…
This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing's theorem states that the edge coloring of a simple graph $G$ requires either $Δ$ or $Δ+1$ colors, where $Δ$ is the maximum vertex degree of $G$. Holyer proved that it is {\bf NP}-complete to decide whether $G$ is $Δ$-edge-colorable even for cubic graphs. By introducing the concept of complex colors, we show that the color-exchange operation follows the same multiplication rules as quaternion. An initially $Δ$-edge-colored graph $G$ allows variable-colored edges, which…
Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $Δ$, it follows from Vizing's Theorem that $O(m\log Δ)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $Δ$ is. On the other h…
Let $G=(V,E)$ be a simple graph of maximum degree $Δ$. The edges of $G$ can be colored with at most $Δ+1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $Δ$ colors. Vizing's Theorem gives a bound of $\fracΔ{Δ+1}|E|$. This is known to be tight for cliques $K_{Δ+1}$ when $Δ$ is even. However, for $Δ=3$ it was improved to $26/31|E|$ by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to $6/7|E|$ by Rizzi [Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309, 2009]. It is tight for …
We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $α$ of the graph, in a worst-case update time of $O(\log^3 n \log α)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log ^2 n \log α)$ for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ out-edges…
