Pub Date : 2026-06-01Epub Date: 2026-01-13DOI: 10.1016/j.disc.2026.114993
Kun Cheng , Chengli Li , Feng Liu
A non-complete graph G is said to be t-tough if for every vertex cut S of G, the ratio of to the number of components of is at least t. A complete graph is said to be t-tough for any . The toughness of the graph G is the maximum value of t such that G is t-tough. A graph G is said to be minimally t-tough if and for every . In 2003, Kriesell conjectured that every minimally 1-tough graph contains a vertex of degree 2. In 2018, Katona and Varga generalized this conjecture, asserting that every minimally t-tough graph contains a vertex of degree . Recently, Zheng and Sun disproved the generalized Kriesell conjecture by constructing a family of 4-regular graphs of even order. They also raised the question of whether there exist other minimally t-tough regular graphs that do not satisfy the generalized Kriesell conjecture. In this paper, we provide an affirmative answer by constructing a family of 4-regular graphs of odd order, as well as a family of 6-regular graphs of order , where .
{"title":"Constructions of minimally t-tough regular graphs","authors":"Kun Cheng , Chengli Li , Feng Liu","doi":"10.1016/j.disc.2026.114993","DOIUrl":"10.1016/j.disc.2026.114993","url":null,"abstract":"<div><div>A non-complete graph <em>G</em> is said to be <em>t</em>-tough if for every vertex cut <em>S</em> of <em>G</em>, the ratio of <span><math><mo>|</mo><mi>S</mi><mo>|</mo></math></span> to the number of components of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> is at least <em>t</em>. A complete graph is said to be <em>t</em>-tough for any <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>. The toughness <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of the graph <em>G</em> is the maximum value of <em>t</em> such that <em>G</em> is <em>t</em>-tough. A graph <em>G</em> is said to be minimally <em>t</em>-tough if <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>t</mi></math></span> and <span><math><mi>τ</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>e</mi><mo>)</mo><mo><</mo><mi>t</mi></math></span> for every <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In 2003, Kriesell conjectured that every minimally 1-tough graph contains a vertex of degree 2. In 2018, Katona and Varga generalized this conjecture, asserting that every minimally <em>t</em>-tough graph contains a vertex of degree <span><math><mo>⌈</mo><mn>2</mn><mi>t</mi><mo>⌉</mo></math></span>. Recently, Zheng and Sun disproved the generalized Kriesell conjecture by constructing a family of 4-regular graphs of even order. They also raised the question of whether there exist other minimally <em>t</em>-tough regular graphs that do not satisfy the generalized Kriesell conjecture. In this paper, we provide an affirmative answer by constructing a family of 4-regular graphs of odd order, as well as a family of 6-regular graphs of order <span><math><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>k</mi><mo>≥</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114993"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980075","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}
The research on cyclic subspace codes aims to design coding schemes with larger cardinality and optimal minimum distance to meet the demands of modern communication systems for efficient and reliable coding. This paper investigates the construction problem of cyclic subspace codes, utilizing combinatorial numbers to select the exponent of the irreducible element γ to construct different Sidon spaces, including k-dimensional and -dimensional spaces. Subsequently, we consider merging the subspace codes generated by these Sidon spaces, resulting in cyclic subspace codes with larger cardinality. Our construction method effectively increases the cardinality of the code while ensuring optimal minimum distance.
{"title":"Construction of large cyclic subspace codes via Sidon spaces with dimensions k and k + 1","authors":"Yongfeng Niu , Chenyu Zhang , Yansheng Wu , Fagang Li","doi":"10.1016/j.disc.2026.115008","DOIUrl":"10.1016/j.disc.2026.115008","url":null,"abstract":"<div><div>The research on cyclic subspace codes aims to design coding schemes with larger cardinality and optimal minimum distance to meet the demands of modern communication systems for efficient and reliable coding. This paper investigates the construction problem of cyclic subspace codes, utilizing combinatorial numbers to select the exponent of the irreducible element <em>γ</em> to construct different Sidon spaces, including <em>k</em>-dimensional and <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional spaces. Subsequently, we consider merging the subspace codes generated by these Sidon spaces, resulting in cyclic subspace codes with larger cardinality. Our construction method effectively increases the cardinality of the code while ensuring optimal minimum distance.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 115008"},"PeriodicalIF":0.7,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980077","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}
Pub Date : 2026-05-01Epub Date: 2026-01-06DOI: 10.1016/j.disc.2025.114974
G. Batta , L. Hajdu
We are concerned with the minimal representation of graphs as common factor graphs. First we show that for any graph G of order k one can find to represent G such that the number of prime divisors of is at most , and that this value is best possible. Then we give upper and lower bounds (which differ only in a constant factor in the exponent) for the smallest n such that every graph of order k is an induced subgraph of the common factor graph induced by the set . Further, we answer a question of Eggleton from 1987 concerning graphs which are extremal for this type of representability to the negative, formulate a conjecture containing three assertions, and provide some related numerical results.
{"title":"Minimal common factor graphs containing all graphs of order k","authors":"G. Batta , L. Hajdu","doi":"10.1016/j.disc.2025.114974","DOIUrl":"10.1016/j.disc.2025.114974","url":null,"abstract":"<div><div>We are concerned with the minimal representation of graphs as common factor graphs. First we show that for any graph <em>G</em> of order <em>k</em> one can find <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi>N</mi></math></span> to represent <em>G</em> such that the number of prime divisors of <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is at most <span><math><mo>⌊</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></math></span>, and that this value is best possible. Then we give upper and lower bounds (which differ only in a constant factor in the exponent) for the smallest <em>n</em> such that every graph of order <em>k</em> is an induced subgraph of the common factor graph induced by the set <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Further, we answer a question of Eggleton from 1987 concerning graphs which are extremal for this type of representability to the negative, formulate a conjecture containing three assertions, and provide some related numerical results.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114974"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939113","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}
Pub Date : 2026-05-01Epub Date: 2026-01-14DOI: 10.1016/j.disc.2026.114977
Yuan Cao , Yonglin Cao , Yanyan Gao , Fanghui Ma , Qin Yue
<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field of <em>q</em> elements. For any integer <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span> be the generalized quaternion group of 4<em>n</em> elements and let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> be the dihedral group of 2<em>n</em> elements. Then left ideals of the group algebra <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub><mo>]</mo></math></span> (resp. <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>]</mo></math></span>) are called left generalized quaternion group codes (resp. left dihedral codes) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of length 4<em>n</em> (resp. 2<em>n</em>) and abbreviated as left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-codes (resp. left <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-codes) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this paper, let <em>q</em> be odd and <span><math><mrow><mi>gcd</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We prove that any left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-code over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is permutation equivalent to a matrix product code by a unique left <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-code and a unique left twisted <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-code over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Then we give a precise representation of left twisted <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and determine all distinct self-dual codes, self-orthogonal codes and linear complementary dual (LCD) codes among these codes. Hence, we obtain a complete enumeration of all distinct self-orthogonal codes and LCD codes among left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. As applications, we correct some mistakes in Gao and Yue (2021) <span><span>[27]</span></spa
{"title":"Matrix product structure of left generalized quaternion group codes","authors":"Yuan Cao , Yonglin Cao , Yanyan Gao , Fanghui Ma , Qin Yue","doi":"10.1016/j.disc.2026.114977","DOIUrl":"10.1016/j.disc.2026.114977","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field of <em>q</em> elements. For any integer <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span> be the generalized quaternion group of 4<em>n</em> elements and let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span> be the dihedral group of 2<em>n</em> elements. Then left ideals of the group algebra <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub><mo>]</mo></math></span> (resp. <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>]</mo></math></span>) are called left generalized quaternion group codes (resp. left dihedral codes) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of length 4<em>n</em> (resp. 2<em>n</em>) and abbreviated as left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-codes (resp. left <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-codes) over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this paper, let <em>q</em> be odd and <span><math><mrow><mi>gcd</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. We prove that any left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-code over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is permutation equivalent to a matrix product code by a unique left <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-code and a unique left twisted <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-code over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Then we give a precise representation of left twisted <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math></span>-codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and determine all distinct self-dual codes, self-orthogonal codes and linear complementary dual (LCD) codes among these codes. Hence, we obtain a complete enumeration of all distinct self-orthogonal codes and LCD codes among left <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math></span>-codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. As applications, we correct some mistakes in Gao and Yue (2021) <span><span>[27]</span></spa","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114977"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977657","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}
Pub Date : 2026-05-01Epub Date: 2026-01-13DOI: 10.1016/j.disc.2026.114981
H.A. Kierstead , Alexandr Kostochka , Zimu Xiang
A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most 1. For a list assignment L of k colors to each vertex of an n-vertex graph G, an equitable L-coloring of G is a proper coloring of vertices of G from their lists such that no color is used more than times. Call a graph equitably k-choosable if it has an equitable L-coloring for every k-list assignment L. A graph G is -sparse if for every , the number of edges in the subgraph of G induced by A is at most .
Our first main result is that every -sparse graph with minimum degree at least 2 is equitably 3-colorable and equitably 3-choosable. This is sharp. Our second main result is that every -sparse graph with minimum degree at least 2 is equitably 4-colorable and equitably 4-choosable. This is also sharp.
One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.
{"title":"Equitable list coloring of sparse graphs","authors":"H.A. Kierstead , Alexandr Kostochka , Zimu Xiang","doi":"10.1016/j.disc.2026.114981","DOIUrl":"10.1016/j.disc.2026.114981","url":null,"abstract":"<div><div>A proper vertex coloring of a graph is <em>equitable</em> if the sizes of all color classes differ by at most 1. For a list assignment <em>L</em> of <em>k</em> colors to each vertex of an <em>n</em>-vertex graph <em>G</em>, an <em>equitable L</em>-coloring of <em>G</em> is a proper coloring of vertices of <em>G</em> from their lists such that no color is used more than <span><math><mo>⌈</mo><mi>n</mi><mo>/</mo><mi>k</mi><mo>⌉</mo></math></span> times. Call a graph <em>equitably k</em>-<em>choosable</em> if it has an equitable <em>L</em>-coloring for every <em>k</em>-list assignment <em>L</em>. A graph <em>G</em> is <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>-<em>sparse</em> if for every <span><math><mi>A</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, the number of edges in the subgraph <span><math><mi>G</mi><mo>[</mo><mi>A</mi><mo>]</mo></math></span> of <em>G</em> induced by <em>A</em> is at most <span><math><mi>a</mi><mo>|</mo><mi>A</mi><mo>|</mo><mo>+</mo><mi>b</mi></math></span>.</div><div>Our first main result is that every <span><math><mo>(</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></math></span>-sparse graph with minimum degree at least 2 is equitably 3-colorable and equitably 3-choosable. This is sharp. Our second main result is that every <span><math><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>-sparse graph with minimum degree at least 2 is equitably 4-colorable and equitably 4-choosable. This is also sharp.</div><div>One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114981"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977835","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}
Pub Date : 2026-05-01Epub Date: 2025-12-17DOI: 10.1016/j.disc.2025.114948
Erfei Yue
<div><div>A family of disjoint pairs of finite sets <span><math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>|</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>m</mi><mo>]</mo><mo>}</mo></math></span> is called a Bollobás system if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>, and a skew Bollobás system if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>i</mi><mo><</mo><mi>j</mi></math></span>. Bollobás proved that for a Bollobás system, the inequality<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn></math></span></span></span> holds. Hegedüs and Frankl proved that for a skew Bollobás system, we have<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>,</mo></math></span></span></span> provided <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span>. In this paper, we improve this inequality to<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi>
{"title":"Some new Bollobás-type inequalities","authors":"Erfei Yue","doi":"10.1016/j.disc.2025.114948","DOIUrl":"10.1016/j.disc.2025.114948","url":null,"abstract":"<div><div>A family of disjoint pairs of finite sets <span><math><mi>P</mi><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>|</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>m</mi><mo>]</mo><mo>}</mo></math></span> is called a Bollobás system if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>, and a skew Bollobás system if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math></span> for every <span><math><mi>i</mi><mo><</mo><mi>j</mi></math></span>. Bollobás proved that for a Bollobás system, the inequality<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn></math></span></span></span> holds. Hegedüs and Frankl proved that for a skew Bollobás system, we have<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>⩽</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>,</mo></math></span></span></span> provided <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊆</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span>. In this paper, we improve this inequality to<span><span><span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msup><mrow><mo>(</mo><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>+</mo><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114948"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798255","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}
Euler's partition identity states that the number of partitions of n into odd parts is equal to the number of partitions of n into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook length (perimeter) n. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.
{"title":"Fixed perimeter analogues of some partition results","authors":"Gabriel Gray , Emily Payne , Holly Swisher , Ren Watson","doi":"10.1016/j.disc.2025.114968","DOIUrl":"10.1016/j.disc.2025.114968","url":null,"abstract":"<div><div>Euler's partition identity states that the number of partitions of <em>n</em> into odd parts is equal to the number of partitions of <em>n</em> into distinct parts. Strikingly, Straub proved in 2016 that this identity also holds when counting partitions of any size with largest hook length (perimeter) <em>n</em>. This has inspired further investigation of partition identities and inequalities in the fixed perimeter setting. Here, we explore fixed perimeter analogues of some well-known partition results inspired by Euler's partition identity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114968"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939118","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}
Pub Date : 2026-05-01Epub Date: 2026-01-05DOI: 10.1016/j.disc.2025.114964
Lapo Cioni , Luca Ferrari , Rebecca Smith
We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously.
In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.
{"title":"Sorting permutations using a pop stack with a bypass","authors":"Lapo Cioni , Luca Ferrari , Rebecca Smith","doi":"10.1016/j.disc.2025.114964","DOIUrl":"10.1016/j.disc.2025.114964","url":null,"abstract":"<div><div>We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously.</div><div>In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114964"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939116","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}
Pub Date : 2026-05-01Epub Date: 2025-12-18DOI: 10.1016/j.disc.2025.114952
John D. LaGrange
Graphs that possess certain properties of zero-divisor graphs are treated as prototypes for algebraic structure. Of particular interest are those in which prime elements exist in the lattice (under inclusion) of arbitrary intersections of neighborhoods. Zero-divisor graphs of commutative rings with identity are shown to be directed unions of such graphs, and more general graphs that have this property are found to be structurally similar to zero-divisor graphs.
{"title":"On graphs that resemble zero-divisor graphs","authors":"John D. LaGrange","doi":"10.1016/j.disc.2025.114952","DOIUrl":"10.1016/j.disc.2025.114952","url":null,"abstract":"<div><div>Graphs that possess certain properties of zero-divisor graphs are treated as prototypes for algebraic structure. Of particular interest are those in which prime elements exist in the lattice (under inclusion) of arbitrary intersections of neighborhoods. Zero-divisor graphs of commutative rings with identity are shown to be directed unions of such graphs, and more general graphs that have this property are found to be structurally similar to zero-divisor graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114952"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798325","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}
Pub Date : 2026-05-01Epub Date: 2025-12-29DOI: 10.1016/j.disc.2025.114962
Yilun Shang
For a simple graph G, let be the fractional chromatic number of G. Graph coloring has a long history in the study of random graph theory and numerous bounds on the chromatic number have been reported for classical random graph models including binomial random graphs and random regular graphs. However, little is known when it comes to the fractional chromatic number except for the trivial bounds that are directly inherited from deterministic graph theory. Let r be a constant in this paper. For the random regular graph model , we show that for a given rational number k with high probability if the degree r is no less than some integer , which is the solution of a minimization problem over the unit interval. For the binomial random graph model , we find that for a given rational number k with high probability if r is greater than a number , which is determined by an optimization problem over a probability measure on the unit interval.
{"title":"The fractional chromatic number of random graphs","authors":"Yilun Shang","doi":"10.1016/j.disc.2025.114962","DOIUrl":"10.1016/j.disc.2025.114962","url":null,"abstract":"<div><div>For a simple graph <em>G</em>, let <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the fractional chromatic number of <em>G</em>. Graph coloring has a long history in the study of random graph theory and numerous bounds on the chromatic number have been reported for classical random graph models including binomial random graphs and random regular graphs. However, little is known when it comes to the fractional chromatic number except for the trivial bounds that are directly inherited from deterministic graph theory. Let <em>r</em> be a constant in this paper. For the random regular graph model <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub></math></span>, we show that <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>k</mi></math></span> for a given rational number <em>k</em> with high probability if the degree <em>r</em> is no less than some integer <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, which is the solution of a minimization problem over the unit interval. For the binomial random graph model <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi><mo>/</mo><mi>n</mi></mrow></msub></math></span>, we find that <span><math><msup><mrow><mi>χ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>r</mi><mo>/</mo><mi>n</mi></mrow></msub><mo>)</mo><mo>≥</mo><mi>k</mi></math></span> for a given rational number <em>k</em> with high probability if <em>r</em> is greater than a number <span><math><msub><mrow><mover><mrow><mi>r</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>k</mi></mrow></msub></math></span>, which is determined by an optimization problem over a probability measure on the unit interval.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 5","pages":"Article 114962"},"PeriodicalIF":0.7,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885056","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}
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