Alex McDonough , Ulrich Reitebuch , Martin Skrodzki
{"title":"Combinatorial and asymptotic results on the neighborhood grid","authors":"Alex McDonough , Ulrich Reitebuch , Martin Skrodzki","doi":"10.1016/j.dam.2025.03.028","DOIUrl":"10.1016/j.dam.2025.03.028","url":null,"abstract":"<div><div>In various application fields, such as fluid-, cell-, or crowd-simulations, spatial data structures are very important. They answer nearest neighbor queries which are instrumental in performing necessary computations for, e.g., taking the next time step in the simulation. Correspondingly, various such data structures have been developed, one being the <em>neighborhood grid</em>.</div><div>In this paper, we consider combinatorial aspects of this data structure. Particularly, we show that an assumption on uniqueness, made in previous works, is not actually satisfied. We extend the notions of the neighborhood grid to arbitrary grid sizes and dimensions and provide two alternative, correct versions of the proof that was broken by the dissatisfied assumption.</div><div>Furthermore, we explore both the uniqueness of certain states of the data structure as well as when the number of these states is maximized. We provide a partial classification by using the hook-length formula for rectangular Young tableaux. Finally, we conjecture how to extend this to all 2-dimensional cases.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 48-64"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhen Ji , Zhiwei Guo , Eddie Cheng , Ralf Klasing , Yaping Mao
{"title":"The distance-edge-monitoring numbers of subdivision graphs","authors":"Zhen Ji , Zhiwei Guo , Eddie Cheng , Ralf Klasing , Yaping Mao","doi":"10.1016/j.dam.2025.03.015","DOIUrl":"10.1016/j.dam.2025.03.015","url":null,"abstract":"<div><div>For a vertex set <span><math><mi>M</mi></math></span> and an edge <span><math><mi>e</mi></math></span> of a connected graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span> be the set of pairs <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></math></span> with <span><math><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow></math></span> and <span><math><mrow><mi>y</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>≠</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>. For a vertex <span><math><mi>x</mi></math></span>, let <span><math><mrow><mi>E</mi><mi>M</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> be the set of edges <span><math><mi>e</mi></math></span> such that there exists a vertex <span><math><mi>v</mi></math></span> satisfying <span><math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><mi>P</mi><mrow><mo>(</mo><mrow><mo>{</mo><mi>x</mi><mo>}</mo></mrow><mo>,</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span> in <span><math><mi>G</mi></math></span>. A set <span><math><mi>M</mi></math></span> of vertices of a graph <span><math><mi>G</mi></math></span> is called a distance-edge-monitoring (DEM for short) set if, for every edge <span><math><mi>e</mi></math></span> of <span><math><mi>G</mi></math></span>, there exist vertices <span><math><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow></math></span> and <span><math><mrow><mi>y</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mi>e</mi></math></span> belongs to all shortest paths between <span><math><mi>x</mi></math></span> and <span><math><mi>y</mi></math></span>. We denote by <span><math><mrow><mo>dem</mo><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> the size of a smallest DEM set of <span><math><mi>G</mi></math></span>. Given a graph <span><math><mi>G</mi></math></span>, subdividing one edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> means removing the edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span>, adding an extra vertex <span><math><mi>w</mi></math></span>, and adding the edges <span><math><mrow><mi>u</mi><mi>w</mi></mrow></math></span> and <span><math><mrow><mi>w</mi><mi>v</mi></mrow></math></span>. The subdivision graph of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is obtained from the graph <span><math><mi>G</mi></math></span> by subdividing all its edges. In this paper, we study the DEM n","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 37-47"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791867","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":"Weakly toll convexity in graph products","authors":"Polona Repolusk","doi":"10.1016/j.dam.2025.03.018","DOIUrl":"10.1016/j.dam.2025.03.018","url":null,"abstract":"<div><div>The exploration of weakly toll convexity is the focus of this investigation. A weakly toll walk is any walk <span><math><mrow><mi>W</mi><mo>:</mo><mi>u</mi><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>v</mi></mrow></math></span> between <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> such that <span><math><mi>u</mi></math></span> is adjacent only to the vertex <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, which can appear more than once in the walk, and <span><math><mi>v</mi></math></span> is adjacent only to the vertex <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, which can appear more than once in the walk. Through an examination of general graphs and an analysis of weakly toll intervals in both lexicographic and (generalized) corona product graphs, precise values of the weakly toll number for these product graphs are obtained. Notably, in both instances, the weakly toll number is constrained to either 2 or 3. Additionally, the determination of the weakly toll number for the Cartesian and the strong product graphs is established through previously established findings in toll convexity theory. Lastly for all graph products examined within our scope, the weakly toll hull number is consistently determined to be 2.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 15-22"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cecily Sahai C., Sampath Kumar S., Arputha Jose T.
{"title":"Four cycle decomposition of λK(n,2)","authors":"Cecily Sahai C., Sampath Kumar S., Arputha Jose T.","doi":"10.1016/j.dam.2025.03.033","DOIUrl":"10.1016/j.dam.2025.03.033","url":null,"abstract":"<div><div>The Kneser graph <span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> is the graph with the <span><math><mi>k</mi></math></span>-subsets of a fixed <span><math><mi>n</mi></math></span>-set as its vertices, with two <span><math><mi>k</mi></math></span>-subsets adjacent if they are disjoint. Given <span><math><mi>n</mi></math></span> and <span><math><mi>k</mi></math></span>, the existence of Hamilton cycle in <span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> was considered by many authors since 1978. Recently, Merino et al. (2023) proved that all connected Kneser graphs <span><math><mrow><mi>K</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> are Hamiltonian. In this paper, we examine the necessary and sufficient conditions for the existence of a 4-cycle decomposition of <span><math><mi>λ</mi></math></span>-fold Kneser graphs <span><math><mrow><mi>λ</mi><mi>K</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>λ</mi></math></span>-fold Bipartite Kneser graphs <span><math><mrow><mi>λ</mi><mi>B</mi><mi>K</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. The main result of this paper may shed some lights on the <span><math><mi>k</mi></math></span>-cycle decomposition problem of Kneser graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 65-70"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791864","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}
Carl Johan Casselgren , Michał Małafiejski , Krzysztof Pastuszak , Petros A. Petrosyan
{"title":"Near-interval edge colorings of graphs","authors":"Carl Johan Casselgren , Michał Małafiejski , Krzysztof Pastuszak , Petros A. Petrosyan","doi":"10.1016/j.dam.2025.03.011","DOIUrl":"10.1016/j.dam.2025.03.011","url":null,"abstract":"<div><div>An interval edge coloring of a graph is a proper edge coloring by integers such that the colors on the edges incident with any vertex form an interval of integers. Not all graphs are interval colorable; a simple counterexample is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. A near-interval coloring is a proper edge coloring of a graph such that the colors on the edges incident with any vertex is either an interval or a <em>near-interval</em>, where the latter is an interval except for one missing integer.</div><div>We prove that all graphs of maximum degree at most 4, and all Class 1 graphs of maximum degree 5 and no vertices of degree 3 are near-interval colorable, thereby improving previous results by Petrosyan et al. (2010). We also consider the problem of near-interval coloring outerplanar graphs. For bipartite graphs, we prove that every such multigraph of maximum degree at most 5 admits a near-interval coloring, and that for every <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>18</mn></mrow></math></span> there is a bipartite graph of maximum degree <span><math><mi>Δ</mi></math></span> with no near-interval coloring. For the case of bipartite multigraphs, we give analogous examples of graphs of maximum degrees <span><math><mi>Δ</mi></math></span> with no near-interval coloring for every <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>15</mn></mrow></math></span>. Finally, we present classes of bipartite multigraphs of maximum degree 6,7 and 8 that admit near-interval colorings.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 23-36"},"PeriodicalIF":1.0,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143791866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Descent generating polynomials for (n−3)- and (n−4)-stack-sortable (pattern-avoiding) permutations","authors":"Philip B. Zhang , Sergey Kitaev","doi":"10.1016/j.dam.2025.03.032","DOIUrl":"10.1016/j.dam.2025.03.032","url":null,"abstract":"<div><div>In this paper, we find distribution of descents over <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow></math></span>- and <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>)</mo></mrow></math></span>-stack-sortable permutations in terms of Eulerian polynomials. Our results generalize the enumeration results by Claesson, Dukes, and Steingrímsson on <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow></math></span>- and <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>)</mo></mrow></math></span>-stack-sortable permutations. Moreover, we find distribution of descents on <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>-, <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow></math></span>- and <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>)</mo></mrow></math></span>-stack-sortable permutations that avoid any given pattern of length 3, which extends known results in the literature on distribution of descents over pattern-avoiding 1- and 2-stack-sortable permutations. Our distribution results also give enumeration of <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>-, <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow></math></span>- and <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo>)</mo></mrow></math></span>-stack-sortable permutations avoiding any pattern of length 3. One of our conjectures links our work to stack-sorting with restricted stacks, and the other conjecture states that 213-avoiding permutations sortable with <span><math><mi>t</mi></math></span> stacks are equinumerous with 321-avoiding permutations sortable with <span><math><mi>t</mi></math></span> stacks for any <span><math><mi>t</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"372 ","pages":"Pages 1-14"},"PeriodicalIF":1.0,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769185","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":"Research problems from the 1st Chinese–Southeasteuropean conference on discrete mathematics and applications","authors":"Vedran Krčadinac , Shenggui Zhang , Liming Xiong , Dragan Stevanović","doi":"10.1016/j.dam.2025.02.037","DOIUrl":"10.1016/j.dam.2025.02.037","url":null,"abstract":"<div><div>This is a collection of open problems related to/presented at the 1st Chinese–Southeasteuropean conference on discrete mathematics and applications that was held at Serbian Academy of Sciences and Arts in Belgrade, Serbia, from June 9–14, 2024.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 99-104"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739099","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}
Mingzu Zhang , Hongxi Liu , Chia-Wei Lee , Weihua Yang
{"title":"Edge isoperimetric method: At least 2/3 of h-extra edge-connectivity of a kind of cube-based graphs concentrates on 2n−1","authors":"Mingzu Zhang , Hongxi Liu , Chia-Wei Lee , Weihua Yang","doi":"10.1016/j.dam.2025.03.019","DOIUrl":"10.1016/j.dam.2025.03.019","url":null,"abstract":"<div><div>The edge isopermetric problem on hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, proposed by Harper in 1964, is to find a vertex subset with cardinality <span><math><mi>m</mi></math></span> in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, such that the edge cut separating any vertex subset with cardinality <span><math><mi>m</mi></math></span> from its complement has minimum size. Since Harper, Lindsey, Bernstein and Hart solved the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by lexicographic order, the edge isoperimetric problem is intimately tied to many-to-many disjoint paths problem. The maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs of order <span><math><mi>h</mi></math></span> in a connected graph <span><math><mi>G</mi></math></span> can be defined by the minimum modified edge-cut, called the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mi>G</mi></math></span>. It is the cardinality of the minimum set of edges in a connected graph <span><math><mi>G</mi></math></span>, if such a set exists, whose deletion disconnects <span><math><mi>G</mi></math></span> and leaves every remaining component with at least <span><math><mi>h</mi></math></span> vertices. The <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are constructed by adding a matching between some pair copies of <span><math><mi>k</mi></math></span> dimensional subcubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>. The distribution of the values of the <span><math><mi>h</mi></math></span>-extra edge-connectivity on a recursive graph is uneven and presents a concentration phenomenon. In this paper, we start with analysing the fractal properties of the optimal solution of the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>. And it is shown that although the members of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are not isomorphic to each other according to different <span><math><mi>k</mi></math></span> where <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, when <span><math><mi>n</mi></math></span> approaches infinity, the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes presents a concentration phenome","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 167-174"},"PeriodicalIF":1.0,"publicationDate":"2025-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143734923","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":"An exact algorithm for the adjacent vertex distinguishing sum edge coloring problem","authors":"Brian Curcio, Isabel Méndez-Díaz, Paula Zabala","doi":"10.1016/j.dam.2025.03.029","DOIUrl":"10.1016/j.dam.2025.03.029","url":null,"abstract":"<div><div>In this work we define the <em>adjacent vertex distinguishing sum edge coloring problem</em>. This problem consists of finding an assignment of colors to the edges of a graph with the following constraints: every pair of adjacent edges must have a different color, and every pair of adjacent vertices must not have the same set of colors assigned to the edges incident to each. The goal is to minimize the sum of the colors in an edge coloring that satisfies these constraints. This problem is a special case of a large family of problems known as <em>graph labeling</em>, which is a widely used and very popular set of tools to build abstract models for problems that arise in everyday life.</div><div>Some variants of <em>graph labeling problems</em> have been successfully addressed with mixed-integer linear programming (MIP) techniques based on a polyhedral characterization of the set of feasible solutions. We use this approach to develop a <em>Branch and Cut</em> algorithm to solve the problem.</div><div>We propose two MIP models that are computationally evaluated to choose the most promising one and continue with a polyhedral study. This analysis aims to characterize valid inequalities that strengthen the formulation in the hope of improving the algorithm’s performance. These inequalities are added on demand as cutting planes using exact and heuristic separation algorithms. Additionally, we considered the use of an initial heuristic and a specific branching strategy.</div><div>The results show that the algorithm developed allows us to solve instances that were unsolvable using general-purpose solvers. Our polyhedral study and the addition of cutting planes have proved to be crucial factors in solving the most challenging instances.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 80-98"},"PeriodicalIF":1.0,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143734795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A simplified graph parameter and its relationship to the modified Randić index","authors":"Dazhi Lin , Tao Wang","doi":"10.1016/j.dam.2025.03.030","DOIUrl":"10.1016/j.dam.2025.03.030","url":null,"abstract":"<div><div>Recently, Lin introduced a new graph parameter <span><math><mi>ξ</mi></math></span> defined by nine properties and established the inequality <span><math><mrow><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the harmonic index of <span><math><mi>G</mi></math></span>. In this note, we simplify the framework by reducing the required properties to five, defining a new parameter <span><math><mi>ζ</mi></math></span>. We prove <span><math><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the modified Randić index, and characterize the equality case. It is known that <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Then <span><math><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Since <span><math><mi>ζ</mi></math></span> needs fewer properties than <span><math><mi>ξ</mi></math></span>, it has a high possibility that many known parameters satisfy the demands of <span><math><mi>ζ</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 60-64"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715403","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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