Some results on the palette index of graphs

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices

Edge-coloring via fixable subgraphs

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $\delta$ and $\Delta$ are the minimum and maximum degrees of $G$ and if $\delta=\Omega(\log n)$, weak splitting can be solved deterministically in time $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log n)\big)$. Further, if $\delta = \Omega(\log\log n)$ and $\Delta\leq 2^{\varepsilon\delta}$, there is a randomized algorithm with time complexity $O\big(\frac{\Delta}{\delta}\cdot\text{poly}(\log\log n)\big)$

Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure

Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence

We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small constant epsilon > 0. In this paper we devise a drastically faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves the previous state-of-the-art {exponentially} in a wide range of Delta, specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In addition, for small values of Delta our deterministic algorithm outperforms all the existing {randomized} algorithms for this problem. On our way to these results we study the {vertex-coloring} problem on the family of graphs with bounded {neighborhood independence}. This is a large family, which strictly includes line graphs of r-hypergraphs for any r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to {general} graphs. Our main technical contribution is a subroutine that computes an O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq Delta