138 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Product structure of graph classes with strongly sublinear separators

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    We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class G\mathcal{G} admits O(n1ϵ)O(n^{1-\epsilon}) separators, then for any fixed δ(0,ϵ)\delta\in(0,\epsilon) every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-depth and a complete graph of size O(n1ϵ+δ)O(n^{1-\epsilon+\delta}). This result holds with δ=0\delta=0 if we allow HH to have tree-depth O(loglogn)O(\log\log n). Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on δ\delta in our results and the above td(H)O(loglogn)\text{td}(H)\in O(\log\log n) bound are both best possible. We prove that nn-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth tt and a complete graph of size O(n1/t)O(n^{1/t}), which is best possible. Finally, we investigate the conjecture that for any hereditary graph class G\mathcal{G} that admits O(n1ϵ)O(n^{1-\epsilon}) separators, every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-width and a complete graph of size O(n1ϵ)O(n^{1-\epsilon}). We prove this for various classes G\mathcal{G} of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial Expansion Classes" which had an error, added section "Lower Bounds", and added a new author; v4: minor revisions and corrections

    HM 32: New Interpretations in Naval History

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    Selected papers from the twenty-first McMullen Naval History Symposium held at the U.S. Naval Academy, 19–20 September 2019.https://digital-commons.usnwc.edu/usnwc-historical-monographs/1031/thumbnail.jp

    Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring

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    The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n ‘bad events’, each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM ’10] also implied an efficient distributed algorithm for the problem, running in O(log2 n) rounds. For instances with low d, this was improved to O(d 2 + logO(1) log n) by Fischer and Ghaffari [DISC ’17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP ’19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari’s algorithm, we improve the round complexity to O( d log d + logO(1) log n). At the other end of the trade-off, we obtain a logO(1) log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first logO(1) log n-round distributed algorithm for the problem of ∆+o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior logo(1) n-round algorithm was known even for 2∆ − 2-edge coloring

    On the choosability of HH-minor-free graphs

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    Given a graph HH, let us denote by fχ(H)f_\chi(H) and f(H)f_\ell(H), respectively, the maximum chromatic number and the maximum list chromatic number of HH-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that fχ(Kt)=t1f_\chi(K_t)=t-1 for every t2t \ge 2. In contrast, for list coloring it is known that 2to(t)f(Kt)O(t(loglogt)6)2t-o(t) \le f_\ell(K_t) \le O(t (\log \log t)^6) and thus, f(Kt)f_\ell(K_t) is bounded away from the conjectured value t1t-1 for fχ(Kt)f_\chi(K_t) by at least a constant factor. The so-called HH-Hadwiger's conjecture, proposed by Seymour, asks to prove that fχ(H)=v(H)1f_\chi(H)=\textsf{v}(H)-1 for a given graph HH (which would be implied by Hadwiger's conjecture). In this paper, we prove several new lower bounds on f(H)f_\ell(H), thus exploring the limits of a list coloring extension of HH-Hadwiger's conjecture. Our main results are: For every ε>0\varepsilon>0 and all sufficiently large graphs HH we have f(H)(1ε)(v(H)+κ(H))f_\ell(H)\ge (1-\varepsilon)(\textsf{v}(H)+\kappa(H)), where κ(H)\kappa(H) denotes the vertex-connectivity of HH. For every ε>0\varepsilon>0 there exists C=C(ε)>0C=C(\varepsilon)>0 such that asymptotically almost every nn-vertex graph HH with Cnlogn\left\lceil C n\log n\right\rceil edges satisfies f(H)(2ε)nf_\ell(H)\ge (2-\varepsilon)n. The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of HH-minor-free graphs is separated from the natural lower bound (v(H)1)(\textsf{v}(H)-1) by a constant factor for all large graphs HH of linear connectivity. The second result tells us that even when HH is a very sparse graph (with an average degree just logarithmic in its order), f(H)f_\ell(H) can still be separated from (v(H)1)(\textsf{v}(H)-1) by a constant factor arbitrarily close to 22. Conceptually these results indicate that the graphs HH for which f(H)f_\ell(H) is close to (v(H)1)(\textsf{v}(H)-1) are typically rather sparse.Comment: 14 page

    On Two problems of defective choosability

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    Given positive integers pkp \ge k, and a non-negative integer dd, we say a graph GG is (k,d,p)(k,d,p)-choosable if for every list assignment LL with L(v)k|L(v)|\geq k for each vV(G)v \in V(G) and vV(G)L(v)p|\bigcup_{v\in V(G)}L(v)| \leq p, there exists an LL-coloring of GG such that each monochromatic subgraph has maximum degree at most dd. In particular, (k,0,k)(k,0,k)-choosable means kk-colorable, (k,0,+)(k,0,+\infty)-choosable means kk-choosable and (k,d,+)(k,d,+\infty)-choosable means dd-defective kk-choosable. This paper proves that there are 1-defective 3-choosable graphs that are not 4-choosable, and for any positive integers k3\ell \geq k \geq 3, and non-negative integer dd, there are (k,d,)(k,d, \ell)-choosable graphs that are not (k,d,+1)(k,d , \ell+1)-choosable. These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27, 4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353], respectively. Our construction of (k,d,)(k,d, \ell)-choosable but not (k,d,+1)(k,d , \ell+1)-choosable graphs generalizes the construction of Kr\'{a}l' and Sgall in [J. Graph Theory 49, 3(2005), 177-186] for the case d=0d=0.Comment: 12 pages, 4 figure

    The grid-minor theorem revisited

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    We prove that for every planar graph XX of treedepth hh, there exists a positive integer cc such that for every XX-minor-free graph GG, there exists a graph HH of treewidth at most f(h)f(h) such that GG is isomorphic to a subgraph of HKcH\boxtimes K_c. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Defective coloring is perfect for minors

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    The defective chromatic number of a graph class is the infimum kk such that there exists an integer dd such that every graph in this class can be partitioned into at most kk induced subgraphs with maximum degree at most dd. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger's conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture

    Efficient Classification of Locally Checkable Problems in Regular Trees

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    We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [Θ(log n), Θ(n)] region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in O(log n) rounds. If not, it is known that the complexity has to be Θ(n^{1/k}) for some k = 1, 2, ..., and in this case the algorithms also output the right value of the exponent k. In rooted trees in the O(log n) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the O(log n) region remains an open question
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