563 research outputs found

    UMSL Bulletin 2023-2024

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    The 2023-2024 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1088/thumbnail.jp

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    UMSL Bulletin 2022-2023

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    The 2022-2023 Bulletin and Course Catalog for the University of Missouri St. Louis.https://irl.umsl.edu/bulletin/1087/thumbnail.jp

    Undergraduate Catalog of Studies, 2022-2023

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    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

    Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

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    Two graphs GG and HH are homomorphism indistinguishable over a class of graphs F\mathcal{F} if for all graphs FFF \in \mathcal{F} the number of homomorphisms from FF to GG is equal to the number of homomorphisms from FF to HH. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl

    Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

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    Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in its (open) neighborhood. Typical examples of graphs with large proper conflict-free chromatic number include graphs with large chromatic number and bipartite graphs isomorphic to the 11-subdivision of graphs with large chromatic number. In this paper, we prove two rough converse statements that hold even in the list-coloring setting. The first is for sparse graphs: for every graph HH, there exists an integer cHc_H such that every graph with no subdivision of HH is (properly) conflict-free cHc_H-choosable. The second applies to dense graphs: every graph with large conflict-free choice number either contains a large complete graph as an odd minor or contains a bipartite induced subgraph that has large conflict-free choice number. These give two incomparable (partial) answers of a question of Caro, Petru\v{s}evski and \v{S}krekovski. We also prove quantitatively better bounds for minor-closed families, implying some known results about proper conflict-free coloring and odd coloring in the literature. Moreover, we prove that every graph with layered treewidth at most ww is (properly) conflict-free (8w1)(8w-1)-choosable. This result applies to (g,k)(g,k)-planar graphs, which are graphs whose coloring problems have attracted attention recently.Comment: Hickingbotham recently independently announced a paper (arXiv:2203.10402) proving a result similar to the ones in this paper. Please see the notes at the end of this paper for details. v2: add results for odd minors, which applies to graphs with unbounded degeneracy, and change the title of the pape

    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

    Spectral extremal results on edge blow-up of graphs

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    The edge blow-up Fp+1F^{p+1} of a graph FF for an integer p2p\geq 2 is obtained by replacing each edge in FF with a Kp+1K_{p+1} containing the edge, where the new vertices of Kp+1K_{p+1} are all distinct. Let ex(n,F)ex(n,F) and spex(n,F)spex(n,F) be the maximum size and maximum spectral radius of an FF-free graph of order nn, respectively. In this paper, we determine the range of spex(n,Fp+1)spex(n,F^{p+1}) when FF is bipartite and the exact value of spex(n,Fp+1)spex(n,F^{p+1}) when FF is non-bipartite for sufficiently large nn, which are the spectral versions of Tur\'{a}n's problems on ex(n,Fp+1)ex(n,F^{p+1}) solved by Yuan [J. Combin. Theory Ser. B 152 (2022) 379--398]. This generalizes several previous results on Fp+1F^{p+1} for FF being a matching, or a star. Additionally, we also give some other interesting results on Fp+1F^{p+1} for FF being a path, a cycle, or a complete graph. To obtain the aforementioned spectral results, we utilize a combination of the spectral version of the Stability Lemma and structural analyses. These approaches and tools give a new exploration of spectral extremal problems on non-bipartite graphs
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