Hans-Joachim Böckenhauer , Juraj Hromkovič , Dennis Komm , Peter Rossmanith , Moritz Stocker
{"title":"A survey of online knapsack problems","authors":"Hans-Joachim Böckenhauer , Juraj Hromkovič , Dennis Komm , Peter Rossmanith , Moritz Stocker","doi":"10.1016/j.dam.2025.08.011","DOIUrl":"10.1016/j.dam.2025.08.011","url":null,"abstract":"<div><div>We survey the current state of research on the knapsack problem in online and semi-online environments. In particular, we summarize what is known about models where different assumptions commonly made in online computation are relaxed: namely that online algorithms do not know the complete instances they are processing; have to make decisions that are irrevocable; and deal with an input chosen by a malicious adversary.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 492-507"},"PeriodicalIF":1.0,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867325","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":"Parallel token swapping for qubit routing","authors":"Ishan Bansal , Oktay Günlük , Richard Shapley","doi":"10.1016/j.dam.2025.08.005","DOIUrl":"10.1016/j.dam.2025.08.005","url":null,"abstract":"<div><div>In this paper we study a combinatorial reconfiguration problem that involves finding an optimal sequence of swaps to move an initial configuration of tokens that are placed on the vertices of a graph to a final desired one. This problem arises as a crucial step in reducing the depth of a quantum circuit when compiling a quantum algorithm. We provide the first known constant factor approximation algorithms for the parallel token swapping problem on graph topologies that are commonly found in modern quantum computers, including cycle graphs, subdivided star graphs, and grid graphs. We also study the so-called stretch factor of a natural lower bound to the problem, which has been shown to be useful when designing heuristics for the qubit routing problem. Finally, we study the colored version of this reconfiguration problem where some tokens share the same color and are considered indistinguishable.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 480-497"},"PeriodicalIF":1.0,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144860816","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":"The orbits of twisted cubes","authors":"Jia-Jie Liu","doi":"10.1016/j.dam.2025.08.028","DOIUrl":"10.1016/j.dam.2025.08.028","url":null,"abstract":"<div><div>Two vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> in a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> are in the same orbit if there exists an automorphism <span><math><mi>ϕ</mi></math></span> of <span><math><mi>G</mi></math></span> such that <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>v</mi></mrow></math></span>. The orbit number of a graph <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the smallest number of orbits that partition <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. All vertex-transitive graphs <span><math><mi>G</mi></math></span> satisfy <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>. Since the <span><math><mi>n</mi></math></span>-dimensional hypercube, denoted by <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, is vertex-transitive, it follows that <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. The twisted cube (Abraham and Padmanabhan, 1991), denoted by <span><math><mrow><mi>T</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, is an interesting variant of the hypercube. In this paper, we prove that <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>T</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> if <span><math><mrow><mi>n</mi><mo>≤</mo><mn>4</mn></mrow></math></span> and <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>T</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow></mrow></msup></mrow></math></span> if <span><math><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 170-176"},"PeriodicalIF":1.0,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144866052","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 total chromatic number of complete multipartite graphs","authors":"Aseem Dalal, B.S. Panda","doi":"10.1016/j.dam.2025.08.027","DOIUrl":"10.1016/j.dam.2025.08.027","url":null,"abstract":"<div><div>In 1995, Hoffman and Rodger conjectured that the total chromatic number <span><math><mrow><msup><mrow><mi>χ</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></math></span> of the complete <span><math><mi>p</mi></math></span>-partite graph <span><math><mrow><mi>K</mi><mo>=</mo><mi>K</mi><mrow><mo>(</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> if and only if <span><math><mrow><mi>K</mi><mo>≠</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>r</mi></mrow></msub></mrow></math></span> and if <span><math><mi>K</mi></math></span> has an even number of vertices then <span><math><mrow><mi>d</mi><mi>e</mi><mi>f</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>K</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is at least the number of parts of odd size. The conjecture is known to be true when <span><math><mi>K</mi></math></span> has odd number of vertices. When <span><math><mi>K</mi></math></span> is even, the problem is quite difficult and is still open with little progress being made. The problem was settled for complete 3-partite graphs by Chew and Yap in 1992, and for complete 4-partite graphs by Dong and Yap in 2000; the difficulty rises manifold with the increase in the number of parts. In 2014, Dalal and Rodger (Graphs and Combinatorics (2015), 1–15) introduced an approach using amalgamations to attack the conjecture and demonstrated its power by settling the problem for complete 5-partite graphs. Their approach required coloring of all the vertices in each part with the same color. However, if the conjecture is true, then for each <span><math><mrow><mn>3</mn><mo>≤</mo><mi>k</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there are complete <span><math><mrow><mn>2</mn><mi>k</mi></mrow></math></span>-partite graphs <span><math><mi>K</mi></math></span> for which any total coloring of <span><math><mi>K</mi></math></span> in which all the vertices in each part are colored the same would require at least <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span> colors, although <span><math><mrow><msup><mrow><mi>χ</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>=</mo><mi>Δ</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>. In this paper, we provide a generalized technique that allows the ver","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 445-458"},"PeriodicalIF":1.0,"publicationDate":"2025-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144841614","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":"Commutative rings with unit graphs of small vertex-arboricity","authors":"A. Mohammadian , S. Sanusha , T. Asir","doi":"10.1016/j.dam.2025.08.024","DOIUrl":"10.1016/j.dam.2025.08.024","url":null,"abstract":"<div><div>This article primarily aims to investigate the arboricity of the unit graph associated with a commutative ring. It provides a characterization, up to isomorphism, of all finite commutative rings whose unit graphs have a vertex-arboricity or an edge-arboricity of at most two.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 140-148"},"PeriodicalIF":1.0,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826769","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}
Shanshan Zhang , Xiumei Wang , Jinjiang Yuan , C.T. Ng , T.C.E. Cheng
{"title":"Even cycles and perfect matchings in planar graphs","authors":"Shanshan Zhang , Xiumei Wang , Jinjiang Yuan , C.T. Ng , T.C.E. Cheng","doi":"10.1016/j.dam.2025.08.013","DOIUrl":"10.1016/j.dam.2025.08.013","url":null,"abstract":"<div><div>Ear decomposition is a powerful tool for the study of the structure of matchings and enumeration of matchings. Lovász showed that a matching covered graph <span><math><mi>G</mi></math></span> has an ear decomposition starting with an arbitrary edge of <span><math><mi>G</mi></math></span>. A graph <span><math><mi>G</mi></math></span> is cycle-nice if, for each even cycle <span><math><mi>C</mi></math></span> in <span><math><mi>G</mi></math></span>, <span><math><mrow><mi>G</mi><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></mrow></math></span> has a perfect matching. A matching covered graph <span><math><mi>G</mi></math></span> has ear decompositions starting with an arbitrary even cycle in <span><math><mi>G</mi></math></span> if and only if <span><math><mi>G</mi></math></span> is a cycle-nice graph. In this paper we show that the only simple cycle-nice 3-connected planar graphs are the odd wheels and the odd prisms. Using this characterization, we show that every cycle-nice matching covered planar graph is an even cycle with multiple edges, or can be obtained from an odd wheel or an odd prism by a sequence of three types of operations.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"378 ","pages":"Pages 468-479"},"PeriodicalIF":1.0,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840732","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":"Generalized paths and cycles in semicomplete multipartite digraphs","authors":"Jørgen Bang-Jensen , Yun Wang , Anders Yeo","doi":"10.1016/j.dam.2025.08.021","DOIUrl":"10.1016/j.dam.2025.08.021","url":null,"abstract":"<div><div>A digraph is <strong>semicomplete</strong> if it has no pair of non-adjacent vertices. It is <strong>complete</strong> if every pair of distinct vertices induces a 2-cycle. A digraph is <strong>semicomplete multipartite</strong> if it can be obtained from a semicomplete digraph <span><math><mi>D</mi></math></span> by choosing a collection of vertex-disjoint subsets <span><math><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span> of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> and then deleting all arcs both of whose end-vertices lie inside some <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. We can also think of a semicomplete digraph as being obtained from a semicomplete multipartite digraph on the same vertex set and partite sets <span><math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span> by adding the arcs of a semicomplete digraph <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> on <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for each partite set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. It is well known that both the hamiltonian path and the hamiltonian cycle problem can be solved in polynomial time for semicomplete multipartite digraphs. In this paper we study the complexity of finding a hamiltonian path or cycle in a semicomplete digraph <span><math><mi>S</mi></math></span> which is obtained as above from a semicomplete multipartite digraph <span><math><mi>D</mi></math></span> and semicomplete digraphs <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>c</mi><mo>]</mo></mrow></mrow></math></span> such that the path or cycle uses as few arcs of <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><mo>…</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span> as possible. We obtain a number of results for the case when each <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a complete digraph. Already this case is highly nontrivial in the cycle case and the complexity is still open. We show how to find a Hamiltonian path which uses as few arcs from the <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>’s as possible in polynomial time and obtain a number of results, both str","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 459-479"},"PeriodicalIF":1.0,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144841613","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":"Self-avoiding closed curves in the regular and semiregular grids","authors":"Lidija Čomić , Paola Magillo","doi":"10.1016/j.dam.2025.08.023","DOIUrl":"10.1016/j.dam.2025.08.023","url":null,"abstract":"<div><div>We consider closed curves in the three regular and eight semiregular grids in the plane, in which each vertex and each edge can be repeated a limited number of times. We define the conditions for such curves to be self-avoiding, and we present a linear-time algorithm to check them. We define the orientation of such curves. We propose a classification of their vertices, and we give a unifying formula relating the number of different types of vertices, valid in the regular and semiregular grids. Our results can be used in the plane tiling applications.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 149-169"},"PeriodicalIF":1.0,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144829058","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}
Amina Riaz , Hafiz Muhammad Afzal Siddiqui , Nasir Ali
{"title":"Graph-theoretic characterization of rings: Outer multiset dimension of zero-divisor graphs","authors":"Amina Riaz , Hafiz Muhammad Afzal Siddiqui , Nasir Ali","doi":"10.1016/j.dam.2025.08.022","DOIUrl":"10.1016/j.dam.2025.08.022","url":null,"abstract":"<div><div>A finite unital commutative ring (UCR) is denoted by <span><math><mi>A</mi></math></span>. The elements <span><math><mrow><mi>ζ</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mi>η</mi><mo>≠</mo><mn>0</mn></mrow></math></span> in <span><math><mi>A</mi></math></span> are zero divisors if their product satisfies <span><math><mrow><mi>ζ</mi><mi>⋅</mi><mi>η</mi><mo>=</mo><mn>0</mn></mrow></math></span>. The set of zero divisor graph in <span><math><mi>A</mi></math></span> is denoted by <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>. A zero divisor graph is constructed using <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span> in order to analyze various algebraic properties. In this article, we characterize the rings based on Outer multiset dimension (OMdim) of their associated zero divisor graphs. For this purpose, we study the zero divisor graphs of rings, including the ring of Gaussian integers modulo <span><math><mi>m</mi></math></span>, <span><math><mrow><msub><mrow><mi>ℨ</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>[</mo><mi>i</mi><mo>]</mo></mrow></mrow></math></span>, the ring of integers modulo <span><math><mi>n</mi></math></span>, <span><math><msub><mrow><mi>ℨ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and certain quotient polynomial rings. Also particularly, we study the OMdim of zero divisor graphs of ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all values of <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 436-444"},"PeriodicalIF":1.0,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827852","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":"Runs in random sequences over ordered sets","authors":"Tanner Reese","doi":"10.1016/j.dam.2025.07.037","DOIUrl":"10.1016/j.dam.2025.07.037","url":null,"abstract":"<div><div>We determine the distributions of lengths of runs in random sequences of elements from a totally ordered set (total order) or partially ordered set (partial order). In particular, we produce novel formulae for the expected value, variance, and probability generating function (PGF) of such lengths in the case of an arbitrary total order. Our focus is on the case of distributions with both atoms and diffuse (absolutely or singularly continuous) mass which has not been addressed in this generality before. We also provide a method of calculating the PGF of run lengths for countably series–parallel partial orders. Additionally, we prove a strong law of large numbers for the distribution of run lengths in a particular realization of an infinite sequence.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"379 ","pages":"Pages 56-78"},"PeriodicalIF":1.0,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144829057","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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