[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81814-en":3,"doc-seo-81814-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81814,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Annihilation, Independence, and Residue: Sharp Matching Bounds for the Annihilation Gap and a TxGraffiti Application","Let G be a finite simple graph. The annihilation number a(G) provides an efficiently computable upper bound on the independence number α(G). The paper develops a sharp matching-number theory for the annihilation gap a(G)−α(G), proving an exact closed-form upper bound in terms of the matching number µ(G) and showing the bound is attainable with prescribed matching size. Sharp matching-dependent results are derived for forests, bipartite graphs, and König–Egerváry graphs, including equality structures and criteria.","arXiv :2607 .01438v1 [math .CO] 1 Jul 2026  \nAnnihilation, Independence, and Residue: Sharp Matching Bounds for the Annihilation Gap  \nand a TxGraffiti Application  \nOhr Kadrawi  \nDepartment of Mathematics Ariel University Ariel 4070000, Israel [orka@ariel.ac.il](orka@ariel.ac.il)  \nVadim E. Levit Department of Mathematics Ariel University Ariel 4070000, Israel [levitv@ariel.ac.il](levitv@ariel.ac.il)  \nAbstract  \nLet G be a finite simple graph. The annihilation number a (G) is an efficiently computable upper bound on the independence number α (G) . We develop a sharp matching-number theory for the gap a (G) − α(G) .  \nThe strongest general theorem is the exact closed forma (G) − α(G) ≤ 2µ(G) + 1 − lp6µ(G)m (µ(G) ≥ 1) ,  \nand the bound is attained for every prescribed matching number. We also prove sharp matching-dependent bounds for forests, bipartite graphs, and König–Egerváry graphs, with equality constructions, equality certificates, and equality criteria.  \nFinally, we treat a TxGraffiti output as a machine-conjecture case study. Using annihilating decompositions together with the classical Havel–Hakimi residue inequality res(G) ≤ α (G), we give an independent proof of the TxGraffiti annihilation-residue inequality  \na (G) + res(G)  \nα (G) ≥  \n∆(G)  \nfor every connected graph G of order at least three, show that both hypotheses are necessary, and compare this proof with a recent Caro– Wei approach. We also refine the Caro–Wei annihilation estimate by an explicit nonnegative slack term, identify its equality cases in degree-sequence form, and combine the refinement with our exact matching-number bound to obtain a combined computable bracket for the independence number and a Gupta–residue bound for the annihilation gap.  \nKeywords—annihilation number, independence number, matching number, König–Egerváry graph, Havel–Hakimi residue, TxGraffiti  \nMathematics Subject Classification (2020)— Primary 05C69; Secondary 05C70, 05C35, 05C85, 68T01 .  \n1 Introduction  \nThroughout the paper all graphs are finite, simple, and undirected. Standard graph terminology follows West [53] . For a graph G, let V (G) and E (G) denote its vertex and edge sets, and put n (G) = |V(G)| and m (G) = |E(G)| . We write dG (v), or simply d (v), for the degree of a vertex v, and ∆(G) for the maximum degree of G.  \nAn independent set is a set of pairwise non-adjacent vertices. The independence number α(G) is the maximum cardinality of an independent set. A matching is a set of pairwise disjoint edges, and the matching number µ (G) is the maximum cardinality of a matching. We write τ (G) = n (G) − α(G) for the vertex cover number; this identity follows because the complement of an independent set is a vertex cover, and conversely. A graph G is called a König–Egerváry graph if  \nα (G) + µ(G) = n(G) .  \nThis terminology and its matching-cover characterizations go back to the work of Deming, Gavril, and Sterboul [14 , 19 , 48] . The classical König–Egerváry theorem for bipartite graphs states that the maximum matching size equals the minimum vertex-cover size; we cite both original papers, by Kőnig and by Egerváry, published in the same 1931 volume of Matematikai és Fizikai Lapok [16 , 28] . Consequently, every bipartite graph is König–Egerváry.  \nLet  \nd 1 ≤ d2 ≤ · · · ≤ dn  \nbe the degree sequence of G. The annihilation number of G, introduced by Pepper [45 , 46], is  \na (G) = max (k : Xi1 di ≤ m(G)) .  \nEquivalently, an annihilating set is a set A ⊆ V (G) such that Pv∈A d (v) ≤ m (G), and a(G) is the maximum size of an annihilating set. This equivalence follows because, among all k-vertex subsets, the sum of the k smallest degrees  \nis the minimum possible degree sum. Thus, an annihilating set need not literally consist of the first k vertices in a degree ordering; the ordered degree sequence merely computes the largest possible cardinality. Every independent set I is annihilating, because each edge of G has at most one endpoint in I , and hence  \nXd(v) = e","cbCaieHdKUG47Qtr","https://ap.wps.com/l/cbCaieHdKUG47Qtr","pdf",693011,5,1,46,"English","en",105,"# Introduction\n## Key graph invariants and definitions\n## Annihilation number and annihilating sets\n## Independence and the annihilation gap","[{\"question\":\"What does the annihilation number a(G) bound and why is it relevant?\",\"answer\":\"For a finite simple graph G, a(G) is an efficiently computable upper bound on the independence number α(G). The paper focuses on how large the gap a(G)−α(G) can be.\"},{\"question\":\"What is the main role of the matching number µ(G) in the results?\",\"answer\":\"The work establishes sharp matching-number bounds for the annihilation gap a(G)−α(G). It gives a strongest general theorem with an exact closed-form inequality and proves the bound can be attained for every prescribed matching number.\"},{\"question\":\"How is the TxGraffiti residue inequality proved and what comparison is made?\",\"answer\":\"The paper treats a TxGraffiti output as a machine-conjecture case study, proving the annihilation–residue inequality using annihilating decompositions and the Havel–Hakimi residue inequality. It also compares this proof with a recent Caro–Wei approach and refines the Caro–Wei estimate with a nonnegative slack term.\"}]","Annihilation, Independence, and Residue: Sharp Matching Bounds for the Annihilation Gap and a TxGraffiti Application | PDF",1784176319,116,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"annihilation-independence-and-residue-sharp-matching-bounds-for-the-annihilation-gap-and-a-txgraffiti-application","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/annihilation-independence-and-residue-sharp-matching-bounds-for-the-annihilation-gap-and-a-txgraffiti-application/81814/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-30","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What does the annihilation number a(G) bound and why is it relevant?","Question",{"text":77,"@type":78},"For a finite simple graph G, a(G) is an efficiently computable upper bound on the independence number α(G). The paper focuses on how large the gap a(G)−α(G) can be.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What is the main role of the matching number µ(G) in the results?",{"text":82,"@type":78},"The work establishes sharp matching-number bounds for the annihilation gap a(G)−α(G). It gives a strongest general theorem with an exact closed-form inequality and proves the bound can be attained for every prescribed matching number.",{"name":84,"@type":75,"acceptedAnswer":85},"How is the TxGraffiti residue inequality proved and what comparison is made?",{"text":86,"@type":78},"The paper treats a TxGraffiti output as a machine-conjecture case study, proving the annihilation–residue inequality using annihilating decompositions and the Havel–Hakimi residue inequality. It also compares this proof with a recent Caro–Wei approach and refines the Caro–Wei estimate with a nonnegative slack term.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]