Combinatorica最新文献

{"title":"Tight Bound on Treedepth in Terms of Pathwidth and Longest Path","authors":"","doi":"10.1007/s00493-023-00077-w","DOIUrl":"https://doi.org/10.1007/s00493-023-00077-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show that every graph with pathwidth strictly less than <em>a</em> that contains no path on <span> <span>(2^b)</span> </span> vertices as a subgraph has treedepth at most 10<em>ab</em>. The bound is best possible up to a constant factor.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740534","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":"Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs","authors":"Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf","doi":"10.1007/s00493-023-00078-9","DOIUrl":"https://doi.org/10.1007/s00493-023-00078-9","url":null,"abstract":"<p>For <span>(0 le t le r)</span> let <i>m</i>(<i>t</i>, <i>r</i>) be the maximum number <i>s</i> such that every <i>t</i>-edge-connected <i>r</i>-graph has <i>s</i> pairwise disjoint perfect matchings. There are only a few values of <i>m</i>(<i>t</i>, <i>r</i>) known, for instance <span>(m(3,3)=m(4,r)=1)</span>, and <span>(m(t,r) le r-2)</span> for all <span>(t not = 5)</span>, and <span>(m(t,r) le r-3)</span> if <i>r</i> is even. We prove that <span>(m(2l,r) le 3l - 6)</span> for every <span>(l ge 3)</span> and <span>(r ge 2 l)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"38 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740588","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 Topological Version of Hedetniemi’s Conjecture for Equivariant Spaces","authors":"Vuong Bui, Hamid Reza Daneshpajouh","doi":"10.1007/s00493-023-00079-8","DOIUrl":"https://doi.org/10.1007/s00493-023-00079-8","url":null,"abstract":"<p>A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two <span>({mathbb {Z}}/2)</span>- spaces is equal to the minimum of their <span>({mathbb {Z}}/2)</span>-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for <i>G</i>-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of <i>G</i>-spaces. More precisely, we show that this conjecture can possibly survive if the group <i>G</i> is either a cyclic <i>p</i>-group or a generalized quaternion group whose size is a power of 2.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138740501","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 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}