Б.Б. Кайдар, Гульнара Танатаровна Смагулова, Э. Брахим, Зулхаир Аймухаметович Мансуров
Одной из основных проблем, с которыми фермы по всему миру сталкиваются – различного вида микотоксикозы у сельскохозяйственных животных. Это заболевания, вызванные токсичными веществами, продуцируемыми различными видами грибов, таких как Fusarium, Aspergillus и Penicillium, которые образовываются на растениях и кормах. В работе представлены общие данные по микотоксинам: основные виды, оказывающие наибольший негативный эффект на организм животных и птиц, а также исследования по применению различных сорбентов в качестве детоксифицирующих агентов. Представлен способ получения активированных углей из биоотхода в виде свекловичного жома.
{"title":"Применение активированных углей в борьбе с микотоксинами","authors":"Б.Б. Кайдар, Гульнара Танатаровна Смагулова, Э. Брахим, Зулхаир Аймухаметович Мансуров","doi":"10.18321/CPC353","DOIUrl":"https://doi.org/10.18321/CPC353","url":null,"abstract":"Одной из основных проблем, с которыми фермы по всему миру сталкиваются – различного вида микотоксикозы у сельскохозяйственных животных. Это заболевания, вызванные токсичными веществами, продуцируемыми различными видами грибов, таких как Fusarium, Aspergillus и Penicillium, которые образовываются на растениях и кормах. В работе представлены общие данные по микотоксинам: основные виды, оказывающие наибольший негативный эффект на организм животных и птиц, а также исследования по применению различных сорбентов в качестве детоксифицирующих агентов. Представлен способ получения активированных углей из биоотхода в виде свекловичного жома.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44693792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Г. Е. Ергазиева, Р.Х. Тургумбаева, К.М. Сахиева, К. Досумов, М. М. Тельбаева
В реакции углекислотной конверсии метана до синтез–газа исследованы однокомпонентные катализаторы на основе оксидов (NiO, Co2O3 и МоО3), нанесенные на различные носители (θ-Al2O3, γ-Al2O3, 5А, 4А, 3А, 13Х, HY и HZSM-5). Определен оптимальный состав катализатора на основе оксида никеля с содержанием на носителе 3 масс.%, оксида кобальта – 15 масс.% и оксида молибдена-10 масс.%. Изучено влияние методов приготовления на активность эффективного 3 масс.% NiO/ γ-Al2O3 катализатора в реакции углекислотной конверсии метана в Н2 и СО. Для никелевого катализатора с низким содержанием активной фазы выбран оптимальный способ приготовления (метод каппилярной пропитки носителя по влагоемкости). На 3 масс.% NiO/ γ-Al2O3 катализаторе при условии реакции СН4:СО2 = 1:1, Тр - 8000С, W=1500 ч-1 значения конверсии и выхода целевых продуктов составляют: ХСН4- 89%, ХСО2 - 93 %, С (Н2)- 45,4 %, С (СО) – 42,4 %.
{"title":"Однокомпанентные катализаторы синтеза смеси Н2 и СО из природного газа метана","authors":"Г. Е. Ергазиева, Р.Х. Тургумбаева, К.М. Сахиева, К. Досумов, М. М. Тельбаева","doi":"10.18321/CPC350","DOIUrl":"https://doi.org/10.18321/CPC350","url":null,"abstract":"В реакции углекислотной конверсии метана до синтез–газа исследованы однокомпонентные катализаторы на основе оксидов (NiO, Co2O3 и МоО3), нанесенные на различные носители (θ-Al2O3, γ-Al2O3, 5А, 4А, 3А, 13Х, HY и HZSM-5). Определен оптимальный состав катализатора на основе оксида никеля с содержанием на носителе 3 масс.%, оксида кобальта – 15 масс.% и оксида молибдена-10 масс.%. Изучено влияние методов приготовления на активность эффективного 3 масс.% NiO/ γ-Al2O3 катализатора в реакции углекислотной конверсии метана в Н2 и СО. Для никелевого катализатора с низким содержанием активной фазы выбран оптимальный способ приготовления (метод каппилярной пропитки носителя по влагоемкости). На 3 масс.% NiO/ γ-Al2O3 катализаторе при условии реакции СН4:СО2 = 1:1, Тр - 8000С, W=1500 ч-1 значения конверсии и выхода целевых продуктов составляют: ХСН4- 89%, ХСО2 - 93 %, С (Н2)- 45,4 %, С (СО) – 42,4 %.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46329477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
А. Т. Канаев, А Гуляренко, П А Тополянский, Т Сарсембаева
Экспериментально показано, что в плазменно-закаленной стали 65Г по глубине упрочненной зоны формируется градиентно-слоистая структура с закономерно изменяющейся дисперсностью и микротвёрдостью структурных составляющих. Причиной мелкодисперсного характера формируемой градиентно-слоистой структуры является сверхвысокая скорость охлаждения, вызывающая высокий градиент температуры вблизи поверхности. Отмечено, что присутствие небольшого количества остаточного аустенита в структуре поверхностного и приповерхностного слоев является положительным фактором, поскольку аустенитные прослойки с повышенной вязкостью по границам мартенситных пластин являются барьерами для распространения трещин из закаленного слоя в основной металл. Показано, что при сверхскоростном нагреве и охлаждении желаемые свойства металла достигаются не в результате выделения равновесных фаз при превращениях, а благодаря формированию тех или иных метастабильных фаз и структур, характеризующихся ярко выраженной химической неоднородностью.
{"title":"Плазменная закалка лемеха плуга из конструкционной стали 65Г","authors":"А. Т. Канаев, А Гуляренко, П А Тополянский, Т Сарсембаева","doi":"10.18321/CPC351","DOIUrl":"https://doi.org/10.18321/CPC351","url":null,"abstract":"Экспериментально показано, что в плазменно-закаленной стали 65Г по глубине упрочненной зоны формируется градиентно-слоистая структура с закономерно изменяющейся дисперсностью и микротвёрдостью структурных составляющих. Причиной мелкодисперсного характера формируемой градиентно-слоистой структуры является сверхвысокая скорость охлаждения, вызывающая высокий градиент температуры вблизи поверхности. Отмечено, что присутствие небольшого количества остаточного аустенита в структуре поверхностного и приповерхностного слоев является положительным фактором, поскольку аустенитные прослойки с повышенной вязкостью по границам мартенситных пластин являются барьерами для распространения трещин из закаленного слоя в основной металл. Показано, что при сверхскоростном нагреве и охлаждении желаемые свойства металла достигаются не в результате выделения равновесных фаз при превращениях, а благодаря формированию тех или иных метастабильных фаз и структур, характеризующихся ярко выраженной химической неоднородностью.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41443664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Владимир Сергеевич Арутюнов, К. Я. Трошин, Андрей Александрович Беляев, Арам Владимирович Арутюнов, А. В. Никитин, Л.Н. Стрекова
Представлен обзор результатов экспериментальных исследований и кинетического анализа самовоспламенения смесей метана с алканами С2–С6, водородом и рядом других газов в диапазоне температур 700–1000 К и давлений 1–15 атм, при которых возможно их нежелательное самовоспламенение в ДВС с искровым зажиганием. Показано, что, несмотря на большое различие в реакционной способности алканов С2–С6 влияние их добавок на задержку воспламенения метана практически одинаково. Для многокомпонентных метан-алкановых смесей задержка самовоспламенения определяется суммарной концентрацией алканов и практически не зависит от ее детального состава. Это позволило получить аналитические зависимости для расчета задержки самовоспламенения сложных метан-алкановых смесей. Показано существенное различие в температурной зависимости задержки самовоспламенения метана, водорода и их смесей, связанное с принципиальными отличиями в механизмах их окисления.
{"title":"Влияние состава газовых смесей на задержку их самовоспламенения и нормальную скорость пламени","authors":"Владимир Сергеевич Арутюнов, К. Я. Трошин, Андрей Александрович Беляев, Арам Владимирович Арутюнов, А. В. Никитин, Л.Н. Стрекова","doi":"10.18321/CPC349","DOIUrl":"https://doi.org/10.18321/CPC349","url":null,"abstract":"Представлен обзор результатов экспериментальных исследований и кинетического анализа самовоспламенения смесей метана с алканами С2–С6, водородом и рядом других газов в диапазоне температур 700–1000 К и давлений 1–15 атм, при которых возможно их нежелательное самовоспламенение в ДВС с искровым зажиганием. Показано, что, несмотря на большое различие в реакционной способности алканов С2–С6 влияние их добавок на задержку воспламенения метана практически одинаково. Для многокомпонентных метан-алкановых смесей задержка самовоспламенения определяется суммарной концентрацией алканов и практически не зависит от ее детального состава. Это позволило получить аналитические зависимости для расчета задержки самовоспламенения сложных метан-алкановых смесей. Показано существенное различие в температурной зависимости задержки самовоспламенения метана, водорода и их смесей, связанное с принципиальными отличиями в механизмах их окисления.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44782497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-19DOI: 10.1017/s0963548320000395
N. Alon, R. Briceño, Nishant Chandgotia, A. Magazinov, Yinon Spinka
We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $qle d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $qge d+2$ , any proper q-colouring of the boundary of a box of side length $n ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $qgeq 2d+1$ , the latter holds for any $n ge 1$ . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.
我们研究并分类了lgd格的固有q-着色，确定了具有不同组合行为的三种区域。(1)当$qle d+1$时，存在冻结着色，即存在不能在任意有限子集上被修改的适q-着色。(2)证明了一个强表着色性质，该性质表明，当$qge d+2$时，边长为$n ge d+2$的盒子的边界的任何适当q-着色都可以推广为整个盒子的适当q-着色。(3)当$qgeq 2d+1$时，后者适用于任何$n ge 1$。因此，我们根据它们的混合性质对素格的固有q色空间进行了分类。
{"title":"Mixing properties of colourings of the ℤd lattice","authors":"N. Alon, R. Briceño, Nishant Chandgotia, A. Magazinov, Yinon Spinka","doi":"10.1017/s0963548320000395","DOIUrl":"https://doi.org/10.1017/s0963548320000395","url":null,"abstract":"We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When \u0000$qle d+1$\u0000, there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when \u0000$qge d+2$\u0000, any proper q-colouring of the boundary of a box of side length \u0000$n ge d+2$\u0000 can be extended to a proper q-colouring of the entire box. (3) When \u0000$qgeq 2d+1$\u0000, the latter holds for any \u0000$n ge 1$\u0000. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75142486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-19DOI: 10.1017/s0963548320000139
Marthe Bonamy, N. Bousquet, G. Perarnau
Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $kgeq Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $kgeq Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.
{"title":"Frozen (Δ + 1)-colourings of bounded degree graphs","authors":"Marthe Bonamy, N. Bousquet, G. Perarnau","doi":"10.1017/s0963548320000139","DOIUrl":"https://doi.org/10.1017/s0963548320000139","url":null,"abstract":"Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for \u0000$kgeq Delta+2$\u0000. Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for \u0000$kgeq Delta+2$\u0000 colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91004180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-15DOI: 10.1017/s0963548320000279
Ararat Harutyunyan, P. Horn, Jacques Verstraëte
Let $gamma(G)$ and $${gamma _ circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then$$begin{equation}gamma(G) leq frac{n}{d}(log d + 1).end{equation}$$In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then$$begin{equation}gamma_(G) leq frac{n}{d}(log d + c)end{equation}$$for some constant c independent of d. This result is sharp in the sense that as $d rightarrow infty$ , almost all d-regular n-vertex graphs G of girth at least five have$$begin{equation}gamma_(G) sim frac{n}{d}log d.end{equation}$$Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs
{"title":"Independent dominating sets in graphs of girth five","authors":"Ararat Harutyunyan, P. Horn, Jacques Verstraëte","doi":"10.1017/s0963548320000279","DOIUrl":"https://doi.org/10.1017/s0963548320000279","url":null,"abstract":"<jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000279_inline1.png\" /><jats:tex-math>\u0000$gamma(G)$\u0000</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000279_inline2.png\" /><jats:tex-math>\u0000$${gamma _ circ }(G)$$\u0000</jats:tex-math></jats:alternatives></jats:inline-formula> denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if <jats:italic>G</jats:italic> is an <jats:italic>n</jats:italic>-vertex graph of minimum degree at least <jats:italic>d</jats:italic>, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548320000279_eqnu1.png\" /><jats:tex-math>$$begin{equation}gamma(G) leq frac{n}{d}(log d + 1).end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>In this paper the main result is that if <jats:italic>G</jats:italic> is any <jats:italic>n</jats:italic>-vertex <jats:italic>d</jats:italic>-regular graph of girth at least five, then<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548320000279_eqnu2.png\" /><jats:tex-math>$$begin{equation}gamma_(G) leq frac{n}{d}(log d + c)end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula>for some constant <jats:italic>c</jats:italic> independent of <jats:italic>d</jats:italic>. This result is sharp in the sense that as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000279_inline3.png\" /><jats:tex-math>\u0000$d rightarrow infty$\u0000</jats:tex-math></jats:alternatives></jats:inline-formula>, almost all <jats:italic>d</jats:italic>-regular <jats:italic>n</jats:italic>-vertex graphs G of girth at least five have<jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" position=\"float\" xlink:href=\"S0963548320000279_eqnu3.png\" /><jats:tex-math>$$begin{equation}gamma_(G) sim frac{n}{d}log d.end{equation}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>Furthermore, if <jats:italic>G</jats:italic> is a disjoint union of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548320000279_inline4.png\" /><jats:tex-math>\u0000${n}/{(2d)}$\u0000</jats:tex-math></jats:alternatives></jats:inline-formula> complete bipartite graphs <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"ht","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86464481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-12DOI: 10.1017/S0963548321000067
N. Dragani'c, Franccois Dross, J. Fox, António Girão, F. Havet, D'aniel Kor'andi, W. Lochet, David Munh'a Correia, A. Scott, B. Sudakov
In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.
{"title":"Powers of paths in tournaments","authors":"N. Dragani'c, Franccois Dross, J. Fox, António Girão, F. Havet, D'aniel Kor'andi, W. Lochet, David Munh'a Correia, A. Scott, B. Sudakov","doi":"10.1017/S0963548321000067","DOIUrl":"https://doi.org/10.1017/S0963548321000067","url":null,"abstract":"\u0000 In this short note we prove that every tournament contains the k-th power of a directed path of linear length. This improves upon recent results of Yuster and of Girão. We also give a complete solution for this problem when k=2, showing that there is always a square of a directed path of length , which is best possible.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81990256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-12DOI: 10.1017/s0963548321000390
M. Axenovich, António Girão, Richard Snyder, Lea Weber
Kostochka and Thomason independently showed that any graph with average degree $Omega(rsqrt{log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Omega(rsqrt{log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $overrightarrow{K}_{!!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $overrightarrow{K}_{!!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $overrightarrow{K}_{!!r}$ minor, and any tournament with minimum out-degree $Omega(rsqrt{log r})$ also contains a strong $overrightarrow{K}_{!!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f;:;mathbb{N} rightarrow mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $overrightarrow{K}_{!!r}$ minor, but such a function exists when considering dichromatic number.
{"title":"Strong complete minors in digraphs","authors":"M. Axenovich, António Girão, Richard Snyder, Lea Weber","doi":"10.1017/s0963548321000390","DOIUrl":"https://doi.org/10.1017/s0963548321000390","url":null,"abstract":"Kostochka and Thomason independently showed that any graph with average degree $Omega(rsqrt{log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Omega(rsqrt{log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $overrightarrow{K}_{!!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $overrightarrow{K}_{!!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $overrightarrow{K}_{!!r}$ minor, and any tournament with minimum out-degree $Omega(rsqrt{log r})$ also contains a strong $overrightarrow{K}_{!!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f;:;mathbb{N} rightarrow mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $overrightarrow{K}_{!!r}$ minor, but such a function exists when considering dichromatic number.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81315903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-10DOI: 10.1017/S0963548321000134
P. Diaconis, R. Graham, Xiaoyu He, Sam Spiro
Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most $m+O(m^{3/4}log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.
{"title":"Card guessing with partial feedback","authors":"P. Diaconis, R. Graham, Xiaoyu He, Sam Spiro","doi":"10.1017/S0963548321000134","DOIUrl":"https://doi.org/10.1017/S0963548321000134","url":null,"abstract":"\u0000 Consider the following experiment: a deck with m copies of n different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of ‘feedback’ is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in n, at most \u0000 \u0000 \u0000 \u0000$m+O(m^{3/4}log m)$\u0000\u0000 \u0000 cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the \u0000 \u0000 \u0000 \u0000$m=2$\u0000\u0000 \u0000 case was open.","PeriodicalId":10513,"journal":{"name":"Combinatorics, Probability & Computing","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91212783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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