Combinatorica最新文献

{"title":"A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups","authors":"Tao Zhang","doi":"10.1007/s00493-023-00076-x","DOIUrl":"https://doi.org/10.1007/s00493-023-00076-x","url":null,"abstract":"<p>Fuglede’s conjecture states that a subset <span>(Omega subseteq mathbb {R}^{n})</span> with positive and finite Lebesgue measure is a spectral set if and only if it tiles <span>(mathbb {R}^{n})</span> by translation. However, this conjecture does not hold in both directions for <span>(mathbb {R}^n)</span>, <span>(nge 3)</span>. While the conjecture remains unsolved in <span>(mathbb {R})</span> and <span>(mathbb {R}^2)</span>, cyclic groups are instrumental in its study within <span>(mathbb {R})</span>. This paper introduces a new tool to study spectral sets in cyclic groups and, in particular, proves that Fuglede’s conjecture holds in <span>(mathbb {Z}_{p^{n}qr})</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"92 22","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138442741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Characterization of Graphs Whose Small Powers of Their Edge Ideals Have a Linear Free Resolution","authors":"Nguyen Cong Minh, Thanh Vu","doi":"10.1007/s00493-023-00074-z","DOIUrl":"https://doi.org/10.1007/s00493-023-00074-z","url":null,"abstract":"<p>Let <i>I</i>(<i>G</i>) be the edge ideal of a simple graph <i>G</i>. We prove that <span>(I(G)^2)</span> has a linear free resolution if and only if <i>G</i> is gap-free and <span>({{,textrm{reg},}}I(G) le 3)</span>. Similarly, we show that <span>(I(G)^3)</span> has a linear free resolution if and only if <i>G</i> is gap-free and <span>({{,textrm{reg},}}I(G) le 4)</span>. We deduce these characterizations by establishing a general formula for the regularity of powers of edge ideals of gap-free graphs <span>({{,textrm{reg},}}(I(G)^s) = max ({{,textrm{reg},}}I(G) + s-1,2s))</span>, for <span>(s =2,3)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"97 30","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138442303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Universal Planar Graphs for the Topological Minor Relation","authors":"Florian Lehner","doi":"10.1007/s00493-023-00073-0","DOIUrl":"https://doi.org/10.1007/s00493-023-00073-0","url":null,"abstract":"<p>Huynh et al. recently showed that a countable graph <i>G</i> which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds if <i>G</i> contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains every locally finite planar graph as a topological minor. This shows that in the above result it is not enough to require that <i>G</i> contains every locally finite planar graph as a topological minor.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 19","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs","authors":"Jan Corsten, Walner Mendonça","doi":"10.1007/s00493-023-00072-1","DOIUrl":"https://doi.org/10.1007/s00493-023-00072-1","url":null,"abstract":"<p>We prove that for all integers <span>(Delta ,r ge 2)</span>, there is a constant <span>(C = C(Delta ,r) &gt;0)</span> such that the following is true for every sequence <span>({mathcal {F}}= {F_1, F_2, ldots })</span> of graphs with <span>(v(F_n) = n)</span> and <span>(Delta (F_n) le Delta )</span>, for each <span>(n in {mathbb {N}})</span>. In every <i>r</i>-edge-coloured <span>(K_n)</span>, there is a collection of at most <i>C</i> monochromatic copies from <span>({mathcal {F}})</span> whose vertex-sets partition <span>(V(K_n))</span>. This makes progress on a conjecture of Grinshpun and Sárközy.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 20","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kempe Equivalent List Colorings","authors":"Daniel W. Cranston, Reem Mahmoud","doi":"10.1007/s00493-023-00063-2","DOIUrl":"https://doi.org/10.1007/s00493-023-00063-2","url":null,"abstract":"<p>An <span>(alpha ,beta )</span>-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors <span>(alpha )</span> and <span>(beta )</span>. Two <i>k</i>-colorings of a graph are <i>k</i>-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than <i>k</i> colors). Las Vergnas and Meyniel showed that if a graph is <span>((k-1))</span>-degenerate, then each pair of its <i>k</i>-colorings are <i>k</i>-Kempe equivalent. Mohar conjectured the same conclusion for connected <i>k</i>-regular graphs. This was proved for <span>(k=3)</span> by Feghali, Johnson, and Paulusma (with a single exception <span>(K_2square ,K_3)</span>, also called the 3-prism) and for <span>(kge 4)</span> by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment <i>L</i> and an <i>L</i>-coloring <span>(varphi )</span>, a Kempe swap is called <i>L</i>-valid for <span>(varphi )</span> if performing the Kempe swap yields another <i>L</i>-coloring. Two <i>L</i>-colorings are called <i>L</i>-equivalent if we can form one from the other by a sequence of <i>L</i>-valid Kempe swaps. Let <i>G</i> be a connected <i>k</i>-regular graph with <span>(kge 3)</span> and <span>(Gne K_{k+1})</span>. We prove that if <i>L</i> is a <i>k</i>-assignment, then all <i>L</i>-colorings are <i>L</i>-equivalent (again excluding only <span>(K_2square ,K_3)</span>). Further, if <span>(Gin {K_{k+1},K_2square ,K_3})</span>, <i>L</i> is a <span>(Delta )</span>-assignment, but <i>L</i> is not identical everywhere, then all <i>L</i>-colorings of <i>G</i> are <i>L</i>-equivalent. When <span>(kge 4)</span>, the proof is completely self-contained, implying an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let <i>H</i> be a graph such that for every degree-assignment <span>(L_H)</span> all <span>(L_H)</span>-colorings are <span>(L_H)</span>-equivalent. If <i>G</i> is a connected graph that contains <i>H</i> as an induced subgraph, then for every degree-assignment <span>(L_G)</span> for <i>G</i> all <span>(L_G)</span>-colorings are <span>(L_G)</span>-equivalent.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"62 9","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138293144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Book Proof of the Middle Levels Theorem","authors":"Torsten Mütze","doi":"10.1007/s00493-023-00070-3","DOIUrl":"https://doi.org/10.1007/s00493-023-00070-3","url":null,"abstract":"<p>We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the <span>((2n+1))</span>-dimensional hypercube induced by all vertices with exactly <i>n</i> or <span>(n+1)</span> many 1s.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"11 6","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Unique Sums in Abelian Groups","authors":"Benjamin Bedert","doi":"10.1007/s00493-023-00069-w","DOIUrl":"https://doi.org/10.1007/s00493-023-00069-w","url":null,"abstract":"<p>Let <i>A</i> be a subset of the cyclic group <span>({textbf{Z}}/p{textbf{Z}})</span> with <i>p</i> prime. It is a well-studied problem to determine how small |<i>A</i>| can be if there is no unique sum in <span>(A+A)</span>, meaning that for every two elements <span>(a_1,a_2in A)</span>, there exist <span>(a_1',a_2'in A)</span> such that <span>(a_1+a_2=a_1'+a_2')</span> and <span>({a_1,a_2}ne {a_1',a_2'})</span>. Let <i>m</i>(<i>p</i>) be the size of a smallest subset of <span>({textbf{Z}}/p{textbf{Z}})</span> with no unique sum. The previous best known bounds are <span>(log p ll m(p)ll sqrt{p})</span>. In this paper we improve both the upper and lower bounds to <span>(omega (p)log p leqslant m(p)ll (log p)^2)</span> for some function <span>(omega (p))</span> which tends to infinity as <span>(prightarrow infty )</span>. In particular, this shows that for any <span>(Bsubset {textbf{Z}}/p{textbf{Z}})</span> of size <span>(|B|&lt;omega (p)log p)</span>, its sumset <span>(B+B)</span> contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"11 16","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sweeps, Polytopes, Oriented Matroids, and Allowable Graphs of Permutations","authors":"Arnau Padrol, Eva Philippe","doi":"10.1007/s00493-023-00062-3","DOIUrl":"https://doi.org/10.1007/s00493-023-00062-3","url":null,"abstract":"<p>A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Sweeps of a point configuration are in bijection with faces of an associated sweep polytope. Mimicking the fact that sweep polytopes are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. Pseudo-sweeps are a generalization of sweeps in which the sweeping hyperplane is allowed to slightly change direction, and that can be extended to arbitrary oriented matroids in terms of cellular strings. We prove that for sweepable oriented matroids, sweep oriented matroids provide a sphere that is a deformation retract of the poset of pseudo-sweeps. This generalizes a property of sweep polytopes (which can be interpreted as monotone path polytopes of zonotopes), and solves a special case of the strong Generalized Baues Problem for cellular strings. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"54 42","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71514519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Structural Theorem for Sets with Few Triangles","authors":"Sam Mansfield, Jonathan Passant","doi":"10.1007/s00493-023-00066-z","DOIUrl":"https://doi.org/10.1007/s00493-023-00066-z","url":null,"abstract":"<p>We show that if a finite point set <span>(Psubseteq {mathbb {R}}^2)</span> has the fewest congruence classes of triangles possible, up to a constant <i>M</i>, then at least one of the following holds.</p><ul>\u0000<li>\u0000<p>There is a <span>(sigma &gt;0)</span> and a line <i>l</i> which contains <span>(Omega (|P|^sigma ))</span> points of <i>P</i>. Further, a positive proportion of <i>P</i> is covered by lines parallel to <i>l</i> each containing <span>(Omega (|P|^sigma ))</span> points of <i>P</i>.</p>\u0000</li>\u0000<li>\u0000<p>There is a circle <span>(gamma )</span> which contains a positive proportion of <i>P</i>.</p>\u0000</li>\u0000</ul><p> This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"11 18","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families","authors":"P. Frankl","doi":"10.1007/s00493-023-00060-5","DOIUrl":"https://doi.org/10.1007/s00493-023-00060-5","url":null,"abstract":"<p>Let <span>(n ge 2k ge 4)</span> be integers, <span>({[n]atopwithdelims ()k})</span> the collection of <i>k</i>-subsets of <span>([n] = {1, ldots , n})</span>. Two families <span>({mathcal {F}}, {mathcal {G}} subset {[n]atopwithdelims ()k})</span> are said to be <i>cross-intersecting</i> if <span>(F cap G ne emptyset )</span> for all <span>(F in {mathcal {F}})</span> and <span>(G in {mathcal {G}})</span>. A family is called non-trivial if the intersection of all its members is empty. The best possible bound <span>(|{mathcal {F}}| + |{mathcal {G}}| le {n atopwithdelims ()k} - 2 {n - katopwithdelims ()k} + {n - 2k atopwithdelims ()k} + 2)</span> is established under the assumption that <span>({mathcal {F}})</span> and <span>({mathcal {G}})</span> are non-trivial and cross-intersecting. For the proof a strengthened version of the so-called <i>shifting technique</i> is introduced. The most general result is Theorem 4.1.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"11 17","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}