Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2\Delta-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $\Delta$, in $O(\log^7 \Delta \log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrt{\Delta}) + O(\log^* n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2\Delta-1)$-edge-coloring to poly$(\log\log n)$ rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank $r$ --- where each hyperedge has at most $r$ vertices --- with $n$ nodes and maximum degree $\Delta$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r } \Delta \log n)$ rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a $((\log \Delta/\varepsilon)^{O(\log (1/\varepsilon))})$-round deterministic algorithm for $(1+\varepsilon)$-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting $\lambda$-arboricity graphs with out-degree at most $(1+\varepsilon)\lambda$, for any constant $\varepsilon>0$, hence partially answering Open Problem 10 of Barenboim and Elkin's book

Spanning Trails and Spanning Trees

There are two major parts in my dissertation. One is based on spanning trail, the other one is comparing spanning tree packing and covering.;The results of the spanning trail in my dissertation are motivated by Thomassen\u27s Conjecture that every 4-connected line graph is hamiltonian. Harary and Nash-Williams showed that the line graph L( G) is hamiltonian if and only if the graph G has a dominating eulerian subgraph. Also, motivated by the Chinese Postman Problem, Boesch et al. introduced supereulerian graphs which contain spanning closed trails. In the spanning trail part of my dissertation, I proved some results based on supereulerian graphs and, a more general case, spanning trails.;Let alpha(G), alpha\u27(G), kappa( G) and kappa\u27(G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. First, we discuss the 3-edge-connected graphs with bounded edge-cuts of size 3, and prove that any 3-edge-connected graph with at most 11 edge cuts of size 3 is supereulerian, which improves Catlin\u27s result. Second, having the idea from Chvatal-Erdos Theorem which states that every graph G with kappa(G) ≥ alpha( G) is hamiltonian, we find families of finite graphs F 1 and F2 such that if a connected graph G satisfies kappa\u27(G) ≥ alpha(G) -- 1 (resp. kappa\u27(G) ≥ 3 and alpha\u27( G) ≤ 7), then G has a spanning closed trail if and only if G is not contractible to a member of F1 (resp. F2). Third, by solving a conjecture posed in [Discrete Math. 306 (2006) 87-98], we prove if G is essentially 4-edge-connected, then for any edge subset X0 ⊆ E(G) with |X0| ≤ 3 and any distinct edges e, e\u27 2 ∈ E(G), G has a spanning ( e, e\u27)-trail containing all edges in X0.;The results on spanning trees in my dissertation concern spanning tree packing and covering. We find a characterization of spanning tree packing and covering based on degree sequence. Let tau(G) be the maximum number of edge-disjoint spanning trees in G, a(G) be the minimum number of spanning trees whose union covers E(G). We prove that, given a graphic sequence d = (d1, d2···dn) (d1 ≥ d2 ≥···≥ dn) and integers k2 ≥ k1 \u3e 0, there exists a simple graph G with degree sequence d satisfying k 1 ≤ tau(G) ≤ a(G) ≤ k2 if and only if dn ≥ k1 and 2k1(n -- 1) ≤ Sigmani =1 di ≤ 2k2( n -- 1 |I| -- 1) + 2Sigma i∈I di, where I = {lcub}i : di \u3c k2{rcub}

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Upper tails and independence polynomials in random graphs

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-\alpha}$ for an explicit $\alpha=\alpha_H>0$, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where $H$ is a clique. Here we extend the latter work to any fixed graph $H$ and determine a function $c_H(\delta)$ such that, for $p$ as above and any fixed $\delta>0$, the upper tail probability is $\exp[-(c_H(\delta)+o(1))n^2 p^\Delta \log(1/p)]$, where $\Delta$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(\delta)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k$ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(\delta)$ is the minimum between $\frac12 \delta^{2/m}$ and the unique positive solution of $P_H(x) = 1+\delta$