Journal of Combinatorial Optimization最新文献

{"title":"A short proof and experimental study of the approximation algorithm for label s-t cut","authors":"Peng Zhang","doi":"10.1007/s10878-026-01463-y","DOIUrl":"https://doi.org/10.1007/s10878-026-01463-y","url":null,"abstract":"","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148894576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Shortest longest-path graph orientations","authors":"Yuichi Asahiro, Jesper Jansson, Avraham A. Melkman, Eiji Miyano, Hirotaka Ono, Quan Xue, Shay Zakov","doi":"10.1007/s10878-026-01438-z","DOIUrl":"https://doi.org/10.1007/s10878-026-01438-z","url":null,"abstract":"We consider a graph orientation problem that can be viewed as a generalization of Minimum Graph Coloring. Our problem takes as input an undirected graph <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$G = (V, E)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> in which every edge <jats:inline-formula> <jats:alternatives> <jats:tex-math>$${u, v} in E$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>}</mml:mo> <mml:mo>∈</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> has two (potentially different and not necessarily positive) weights representing the lengths of its two possible directions ( <jats:italic>u</jats:italic> , <jats:italic>v</jats:italic> ) and ( <jats:italic>v</jats:italic> , <jats:italic>u</jats:italic> ), and asks for an orientation, i.e., an assignment of a direction to each edge of <jats:italic>G</jats:italic> , such that the length of a longest simple directed path in the resulting directed graph is minimized. A longest path in a graph is not always a maximal path when some edges have negative lengths, so the problem has two variants depending on whether <jats:italic>all</jats:italic> simple directed paths or maximal simple directed paths <jats:italic>only</jats:italic> are taken into account in the definition. We prove that the problems are NP-hard to approximate even if restricted to subcubic planar graphs, and develop fast polynomial-time algorithms for both problem variants for three classes of graphs: path graphs, cycle graphs, and star graphs.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"18 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148883784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perfect hierarchical matchings in graphs","authors":"Ian G. Ludden, Ellen Veomett, Sheldon H. Jacobson","doi":"10.1007/s10878-026-01458-9","DOIUrl":"https://doi.org/10.1007/s10878-026-01458-9","url":null,"abstract":"","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148842960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uncertainty-aware multi-objective optimization for rebalancing bike shared systems","authors":"Diego Daniel Pedroza-Perez, Gabriel Luque, Sergio Nesmachnow, Jamal Toutouh","doi":"10.1007/s10878-026-01460-1","DOIUrl":"https://doi.org/10.1007/s10878-026-01460-1","url":null,"abstract":"Overnight rebalancing in dock-based bike-sharing systems requires routing a limited fleet of trucks before user activity begins. This article formulates static rebalancing problem under stochastic demand uncertainty as a bi-objective combinatorial optimization problem that selects truck routes and visited stations. The first objective minimizes total travel distance. The second objective minimizes scenario-weighted unmet demand under a finite set of demand scenarios derived from historical station-status data. A deterministic recourse evaluation simulates truck loads and station inventories along each route and computes unmet demand for visited and unvisited stations. The article applies two multi-objective evolutionary algorithms, NSGA-II and MOEA/D, using a permutation–partition encoding and relocate-based operators that implement a 1–0 relocate neighborhood between routes. A roulette-wheel-based relocation operator (BB2) biases move selection by the induced change in route distance. Experiments on the Barcelona Bicing network with 518 stations and on clustered subinstances show that NSGA-II attains higher hypervolume and larger non-dominated sets, whereas MOEA/D attains lower runtime; an ablation analysis shows that BB2 improves coverage and proximity indicators.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148842944","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maximum linear arrangement: exact algorithms for specific classes of graphs and approximation algorithms for wide classes of graphs","authors":"Lluís Alemany-Puig, Juan Luis Esteban, Ramon Ferrer-i-Cancho","doi":"10.1007/s10878-026-01454-z","DOIUrl":"https://doi.org/10.1007/s10878-026-01454-z","url":null,"abstract":"Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems. A linear arrangement is usually defined as a permutation of the &lt;jats:italic&gt;n&lt;/jats:italic&gt; vertices of a graph. An intuitive geometric setting is that of vertices lying on consecutive integer positions in the real line, starting at 1; edges are often drawn as semicircles above the real line. A well-known computational problem is the Minimum Linear Arrangement Problem () where the goal is to find an arrangement that minimizes the sum of edge lengths. In this paper we study the Maximum Linear Arrangement problem (), the counterpart of . We devise a new characterization of maximum arrangements of general graphs, and prove that can be solved for &lt;jats:italic&gt;k&lt;/jats:italic&gt; -regular graphs ( &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$kle 2$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;mml:mo&gt;≤&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; ) in time &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$O{left( nright) }$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mfenced&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mfenced&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; , and for &lt;jats:italic&gt;k&lt;/jats:italic&gt; -linear trees ( &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$kle 2$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;mml:mo&gt;≤&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; ) in time &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$O{left( nright) }$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mfenced&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mfenced&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; . We present two constrained variants of we call and . We prove that the former can be solved in time &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$O{left( nright) }$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mfenced&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mfenced&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; for any connected bipartite graph; the latter can be solved by an algorithm that typically runs in time &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:tex-math&gt;$$O{left( n^3log nright) }$$&lt;/jats:tex-math&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;O&lt;/mml:mi&gt; &lt;mml:mfenced&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;log&lt;/mml:mo&gt; &lt;mml:mi&gt;n&lt;/mml:mi&gt; &lt;/mml:mfenced&gt; &lt;/mml:mrow&gt; &lt;/mml:math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; on unlabeled trees. We show that is a 3/2-approximation","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Branch and price algorithm for the stop number minimization problem","authors":"Victor Nascimento, Luidi Simonetti","doi":"10.1007/s10878-026-01459-8","DOIUrl":"https://doi.org/10.1007/s10878-026-01459-8","url":null,"abstract":"The Stop Number Minimization Problem (SNMP), inspired by an autonomous vehicle service from France, arises when a homogeneous fleet of autonomous vehicles transports cargo and personnel across a circuit of stations. The objective is to satisfy all client requests while minimizing the total number of pickup/dropoff stops made by the vehicles, without violating their maximum capacity. We modeled the SNMP as a Pickup and Delivery Problem on an expanded graph with an ordered structure and solved it via a branch-and-price algorithm. The pricing problem was solved using a labeling algorithm with two distinct dominance rules. Additionally, three types of path elimination were introduced in the pricing algorithm. When branching, two strategies were implemented: SOS and Ryan-Foster. We tested our approach in a setting where each vehicle may perform up to 4 laps on the circuit. Computational experiments indicate that this approach solves instances of equivalent size to those tested in previous works in less time. Furthermore, our approach solved instances with up to 80 requests in the single-lap setting, contrasting previous works that solved instances with up to 50 requests.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"34 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148768974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On finding all roots to systems of nonlinear equations using large language models","authors":"Alexander Semenov, Michael J. Hirsch, Panos M. Pardalos","doi":"10.1007/s10878-026-01451-2","DOIUrl":"https://doi.org/10.1007/s10878-026-01451-2","url":null,"abstract":"","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"35 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148690805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Routing among Convex Polygonal Obstacles in the Plane using a Simplified Domain","authors":"Siddharth Gaur, R. Inkulu","doi":"10.1007/s10878-026-01443-2","DOIUrl":"https://doi.org/10.1007/s10878-026-01443-2","url":null,"abstract":"Given a polygonal domain &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal P$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; comprising &lt;italic&gt;h&lt;/italic&gt; pairwise disjoint convex polygonal obstacles together defined with &lt;italic&gt;n&lt;/italic&gt; vertices and a positive real number &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;ϵ&lt;/mml:mi&gt;&lt;mml:mo&gt;&lt;&lt;/mml:mo&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$epsilon &lt; 1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, this paper presents an algorithm to preprocess &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal P$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; to compute routing tables at the vertices of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal P$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; so that a data packet from any vertex of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal P$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is routed to any other vertex of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathcal P$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. At every vertex &lt;italic&gt;v&lt;/italic&gt; of &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mi mathvariant=\"script\"&gt;P&lt;/mml:mi&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} u","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"29 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148715902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two approximation algorithms for solving the min–max heterogeneous weighted delivery problem","authors":"Jianping Li, Ping Yang, Junran Lichen, Runtao Xie","doi":"10.1007/s10878-026-01450-3","DOIUrl":"https://doi.org/10.1007/s10878-026-01450-3","url":null,"abstract":"In this paper, we consider the min–max heterogeneous weighted delivery (MMHWD) problem. Concretely, given a weighted graph &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;G&lt;/mml:mi&gt;&lt;mml:mo&gt;=&lt;/mml:mo&gt;&lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;mml:mo&gt;,&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;mml:mo&gt;;&lt;/mml:mo&gt;&lt;mml:mi&gt;w&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$G=(V,E;w)$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with length function &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;w&lt;/mml:mi&gt;&lt;mml:mo&gt;:&lt;/mml:mo&gt;&lt;mml:mi&gt;E&lt;/mml:mi&gt;&lt;mml:mo stretchy=\"false\"&gt;→&lt;/mml:mo&gt;&lt;mml:msup&gt;&lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;&lt;mml:mo&gt;+&lt;/mml:mo&gt;&lt;/mml:msup&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$w:Erightarrow mathbb {R^{+}}$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; satisfying the triangle inequality, a fixed depot &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mrow&gt;&lt;mml:mi&gt;r&lt;/mml:mi&gt;&lt;mml:mo&gt;∈&lt;/mml:mo&gt;&lt;mml:mi&gt;V&lt;/mml:mi&gt;&lt;/mml:mrow&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$rin V$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, &lt;italic&gt;m&lt;/italic&gt; items and &lt;italic&gt;k&lt;/italic&gt; vehicles having nonuniform speeds &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;mml:mn&gt;1&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda _1$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:msub&gt;&lt;mml:mi&gt;λ&lt;/mml:mi&gt;&lt;mml:mn&gt;2&lt;/mml:mn&gt;&lt;/mml:msub&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda _2$$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;alternatives&gt;&lt;mml:math&gt;&lt;mml:mo&gt;…&lt;/mml:mo&gt;&lt;/mml:math&gt;&lt;tex-math&gt;documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ldots $$end{document}&lt;/tex-math&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, &lt;","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"361 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148715900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Open problems on Euler (elliptic) Sombor index of graphs","authors":"Sneha Sekar, Kinkar Chandra Das, Selvaraj Balachandran, Suresh Elumalai","doi":"10.1007/s10878-026-01456-x","DOIUrl":"https://doi.org/10.1007/s10878-026-01456-x","url":null,"abstract":"Motivated by the geometric framework underlying the Sombor index, the elliptic Sombor index (<italic>ESO</italic>) and the Euler Sombor index (<italic>EU</italic>) have recently been introduced, each arising from a distinct geometric rationale. In this paper, we resolve several open problems and demonstrate the effectiveness of these methods in addressing questions that have remained unsolved until now. We first investigate the extremal properties of the Euler–Sombor index. Specifically, we determine the second-minimal Euler Sombor index among all trees and, based on this result, establish the minimum Euler Sombor index for graphs with a given clique number, along with a characterization of the extremal graphs attaining this bound. Furthermore, we characterize the graphs that achieve the maximum Euler Sombor index among all graphs with a fixed chromatic number. These findings resolve the open problems posed by Kızılırmak (<xref ref-type=\"bibr\">2025a</xref>). We then turn to the study of the elliptic Sombor index. Recently, Ahmad et al. (<xref ref-type=\"bibr\">2025</xref>) investigated the extremal properties of the elliptic Sombor index for trees under specific structural constraints and provided a complete characterization. They also discussed its potential applications in chemical graph theory and posed an open problem: to determine the chemical trees with the maximum elliptic Sombor index among all chemical trees with (<italic>n</italic>) vertices and (<italic>s</italic>) segments. In this work, we completely resolve this open problem. Our results provide new insights into the structural behavior of the elliptic Sombor index in chemical trees and contribute to its further development within the framework of chemical graph theory.","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"7 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148715891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}