{"title":"Coloring zonotopal quadrangulations of the projective space","authors":"Masahiro Hachimori , Atsuhiro Nakamoto , Kenta Ozeki","doi":"10.1016/j.ejc.2024.104089","DOIUrl":"10.1016/j.ejc.2024.104089","url":null,"abstract":"<div><div>A quadrangulation on a surface <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a map of a simple graph on <span><math><msup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> such that each 2-dimensional face is quadrangular. Youngs proved that every quadrangulation on the projective plane <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is either bipartite or 4-chromatic. It is a surprising result since every quadrangulation on an orientable surface with sufficiently high edge-width is 3-colorable. Kaiser and Stehlík defined a <span><math><mi>d</mi></math></span>-dimensional quadrangulation on the <span><math><mi>d</mi></math></span>-dimensional projective space <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for any <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and proved that any such quadrangulation has chromatic number at least <span><math><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></math></span> if it is not bipartite. In this paper, we define another kind of <span><math><mi>d</mi></math></span>-dimensional quadrangulations of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for any <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, and prove that such a quadrangulation <span><math><mi>Q</mi></math></span> is always 4-chromatic if <span><math><mi>Q</mi></math></span> is non-bipartite and satisfies a special geometric condition related to a zonotopal tiling of a <span><math><mi>d</mi></math></span>-dimensional zonotope.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104089"},"PeriodicalIF":1.0,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rectangulotopes","authors":"Jean Cardinal , Vincent Pilaud","doi":"10.1016/j.ejc.2024.104090","DOIUrl":"10.1016/j.ejc.2024.104090","url":null,"abstract":"<div><div>Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional polytopes associated with two combinatorial families of rectangulations composed of <span><math><mi>n</mi></math></span> rectangles. They are defined as quotientopes of natural lattice congruences on the weak Bruhat order on permutations in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and their skeleta are flip graphs on rectangulations. We give simple vertex and facet descriptions of these polytopes, in particular elementary formulas for computing the coordinates of the vertex corresponding to each rectangulation, in the spirit of J.-L. Loday’s realization of the associahedron.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"125 ","pages":"Article 104090"},"PeriodicalIF":1.0,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142756585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the order of semiregular automorphisms of cubic vertex-transitive graphs","authors":"Marco Barbieri , Valentina Grazian , Pablo Spiga","doi":"10.1016/j.ejc.2024.104091","DOIUrl":"10.1016/j.ejc.2024.104091","url":null,"abstract":"<div><div>We prove that, if <span><math><mi>Γ</mi></math></span> is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of <span><math><mi>Γ</mi></math></span> of order at least 6, or the number of vertices of <span><math><mi>Γ</mi></math></span> is bounded above by an absolute constant.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104091"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"More on rainbow cliques in edge-colored graphs","authors":"Xiao-Chuan Liu , Danni Peng , Xu Yang","doi":"10.1016/j.ejc.2024.104088","DOIUrl":"10.1016/j.ejc.2024.104088","url":null,"abstract":"<div><div>In an edge-colored graph <span><math><mi>G</mi></math></span>, a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a complete subgraph on <span><math><mi>k</mi></math></span> vertices in which all the edges have distinct colors. Let <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of edges and colors in <span><math><mi>G</mi></math></span>, respectively. In this paper, we show that for any <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>ɛ</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, then for sufficiently large <span><math><mi>n</mi></math></span>, the number of rainbow cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mi>G</mi></math></span> is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div><div>We also characterize the extremal graphs <span><math><mi>G</mi></math></span> without a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></math></span>, when <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is maximum.</div><div>Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104088"},"PeriodicalIF":1.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"When (signless) Laplacian coefficients meet matchings of subdivision","authors":"Zhibin Du","doi":"10.1016/j.ejc.2024.104087","DOIUrl":"10.1016/j.ejc.2024.104087","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph, whose subdivision is denoted by <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> be the characteristic polynomial of the Laplacian matrix of <span><math><mi>G</mi></math></span>. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of <span><math><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>L</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>, in terms of the spanning forests of <span><math><mi>G</mi></math></span>. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104087"},"PeriodicalIF":1.0,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Freehedra are short","authors":"Daria Poliakova","doi":"10.1016/j.ejc.2024.104084","DOIUrl":"10.1016/j.ejc.2024.104084","url":null,"abstract":"<div><div>We prove the combinatorial property of shortness for freehedra. Note that associahedra, a related family of polytopes, are not short.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104084"},"PeriodicalIF":1.0,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Erdős–Tuza–Valtr conjecture","authors":"Jineon Baek","doi":"10.1016/j.ejc.2024.104085","DOIUrl":"10.1016/j.ejc.2024.104085","url":null,"abstract":"<div><div>The Erdős–Szekeres conjecture states that any set of more than <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></math></span> points in the plane with no three on a line contains the vertices of a convex <span><math><mi>n</mi></math></span>-gon. Erdős, Tuza, and Valtr strengthened the conjecture by stating that any set of more than <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>b</mi></mrow><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>i</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> points in a plane either contains the vertices of a convex <span><math><mi>n</mi></math></span>-gon, <span><math><mi>a</mi></math></span> points lying on a concave downward curve, or <span><math><mi>b</mi></math></span> points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erdős–Szekeres conjecture. We prove the first new case of the Erdős–Tuza–Valtr conjecture since the original 1935 paper of Erdős and Szekeres. Namely, we show that any set of <span><math><mrow><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn></mrow></math></span> points in the plane with no three points on a line and no two points sharing the same <span><math><mi>x</mi></math></span>-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex <span><math><mi>n</mi></math></span>-gon. The proof is also formalized in <em>Lean 4</em>, a computer proof assistance, to ensure the correctness of the proof.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104085"},"PeriodicalIF":1.0,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A combinatorial PROP for bialgebras","authors":"Jorge Becerra","doi":"10.1016/j.ejc.2024.104086","DOIUrl":"10.1016/j.ejc.2024.104086","url":null,"abstract":"<div><div>It is a classical result that the category of finitely-generated free monoids serves as a PROP for commutative bialgebras. Attaching permutations to fix the order of multiplication, we construct an extension of this category that is equivalent to the PROP for bialgebras.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104086"},"PeriodicalIF":1.0,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Signed Mahonian polynomials on derangements in classical Weyl groups","authors":"Kathy Q. Ji , Dax T.X. Zhang","doi":"10.1016/j.ejc.2024.104083","DOIUrl":"10.1016/j.ejc.2024.104083","url":null,"abstract":"<div><div>The polynomial of the major index <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> over the subset <span><math><mi>T</mi></math></span> of the Coxeter group <span><math><mi>W</mi></math></span> is called the Mahonian polynomial over <span><math><mi>T</mi></math></span>, where <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> is a Mahonian statistic of an element <span><math><mrow><mi>σ</mi><mo>∈</mo><mi>T</mi></mrow></math></span>, whereas the polynomial of the major index <span><math><mrow><msub><mrow><mi>maj</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> with the sign <span><math><msup><mrow><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></msup></math></span> over the subset <span><math><mi>T</mi></math></span> is referred to as the signed Mahonian polynomial over <span><math><mi>T</mi></math></span>, where <span><math><mrow><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>W</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> is the length of <span><math><mrow><mi>σ</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. Gessel, Wachs, and Chow established formulas for the Mahonian polynomials over the sets of derangements in the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the hyperoctahedral group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By extending Wachs’ approach and employing a refinement of Stanley’s shuffle theorem established in our recent paper (Ji and Zhang, 2024), we derive a formula for the Mahonian polynomials over the set of derangements in the even-signed permutation group <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. This completes a picture which is now known for all the classical Weyl groups. Gessel–Simion, Adin–Gessel–Roichman, and Biagioli previously established formulas for the signed Mahonian polynomials over the classical Weyl groups. Building upon their formulas, we derive some new formulas for the signed Mahonian polynomials over the set of derangements in classical Weyl groups. As applications of the formulas for the (signed) Mahonian polynomials over the sets of derangements in the classical Weyl groups, we obtain enumerative formulas of the number of derangements in classical Weyl groups with even lengths.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104083"},"PeriodicalIF":1.0,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142577928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lucas Aragão , João Pedro Marciano , Walner Mendonça
{"title":"Degree conditions for Ramsey goodness of paths","authors":"Lucas Aragão , João Pedro Marciano , Walner Mendonça","doi":"10.1016/j.ejc.2024.104082","DOIUrl":"10.1016/j.ejc.2024.104082","url":null,"abstract":"<div><div>A classical result of Chvátal implies that if <span><math><mrow><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>, then any colouring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in red and blue contains either a monochromatic red <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> or a monochromatic blue <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>. We study a natural generalisation of his result, determining the exact minimum degree condition for a graph <span><math><mi>G</mi></math></span> on <span><math><mrow><mi>n</mi><mo>=</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> vertices which guarantees that the same Ramsey property holds in <span><math><mi>G</mi></math></span>. In particular, using a slight generalisation of a result of Haxell, we show that <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mfenced><mrow><mi>t</mi><mo>/</mo><mn>2</mn></mrow></mfenced></mrow></math></span> suffices, and that this bound is best possible. We also use a classical result of Bollobás, Erdős, and Straus to prove a tight minimum degree condition in the case <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>t</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104082"},"PeriodicalIF":1.0,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
