[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81658-en":3,"doc-seo-81658-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},81658,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Exponential Lower Bounds for the Pfaffian Number of Graphs","The Fisher–Kasteleyn–Temperley (FKT) algorithm counts perfect matchings in planar graphs via a single Pfaffian computation, while Galluccio–Loebl and Tesler express the perfect-matching polynomial on orientable surfaces of genus g as a linear combination of at most 4g Pfaffians. This exponential dependence on g is proved unavoidable: for every g ≥ 1, some graph of orientable genus ≤ g needs at least (8/3)g Pfaffians in any such representation, with connected cubic bipartite matching-covered examples. Exponential lower bounds are also established for complete bipartite graphs, improving prior work.","Exponential Lower Bounds for the Pfaffian Number of Graphs  \nPriyanshu Pant  \nIndian Institute of Technology Indore [priyanshupant03@gmail. com](priyanshupant03@gmail. com)  \nRanveer Singh  \nIndian Institute of Technology Indore [ranveer@iiti. ac. in](ranveer@iiti. ac. in)  \narXiv :2605 .21077v2 [math .CO] 9 Jul 2026  \nAbstract  \nThe Fisher–Kasteleyn–Temperley (FKT) algorithm counts perfect matchings in planar graphsin polynomial time using a single Pfaffian computation. Galluccio–Loebl and Tesler extended this Pfaffian method to graphs embedded in an orientable surface of genus g, showing that the perfect-matching polynomial can be written as a linear combination of at most 4g Pfaffians. We prove that this exponential dependence on g is unavoidable in general. More precisely, for every g ≥ 1, there exists a graph of orientable genus at most g whose perfect-matching polynomial requires at least (8/3)g Pfaffians in any such linear representation. In particular, for every even integer n ≥ 6, there is a graph on n vertices with Pfaffian number at least (8/3)⌊n/6⌋ . Moreover, the lower bound is witnessed even by connected cubic bipartite matching-covered graphs of orientable genus exactly g. We also prove exponential lower bounds for complete bipartite graphs, and hence for even complete graphs, improving asymptotically on a recent linear lower bound of Junchaya, Miranda, and Lucchesi.  \n1 Introduction  \nAll graphs considered in this paper are finite and simple, that is, they have no loops or multiple edges. For a graph G, we write V (G) and E (G) for its vertex set and edge set, respectively. A perfect matching of G is a set M of pairwise disjoint edges such that each vertex of G is incident with exactly one edge of M. Counting perfect matchings is a classical problem in graph theory and combinatorics, with applications in chemistry, statistical physics, and quantum mechanics; see [12, 13] . Computationally, the problem is hard. Even for bipartite graphs, it is equivalent to computing the permanent of a 0-1 matrix, which is \\#P-complete by Valiant’s theorem [22] . For an n × n matrix A = (aij), the permanent of A is defined as  \nn  \nper(A) = X Y ai,σ(i) ,  \nσ∈Sn i=1  \nwhere Sn is the set of all permutations of {1, 2 ,..., n}, whereas the determinant is defined by  \nn  \ndet(A) = X sign(σ) Yai,σ(i) ,  \nσ∈Sn i=1  \nwhere sign(σ) is 1 for even permutations and −1 for odd permutations. Although the permanent and determinant differ only in the signs of the same monomials, their computational behavior is strikingly different. The determinant can be computed in O (nω ) arithmetic operations using fast matrix multiplication, where 2 ≤ ω \u003C 2.373 [1, 2] . This contrast in computational complexity  \nmotivates P´olya’s permanent problem [18], which asks when, for a given 0-1 matrix A, one can change signs of the nonzero entries to obtain a matrix B with the same zero pattern such that  \ndet(B) = per(A) .  \nThus, whenever such a signing exists, the permanent of A is computable in polynomial time. This signing problem is equivalent to several combinatorial and graph-theoretic problems, including the existence of Pfaffian orientations for bipartite graphs and the even directed cycle problem [14, 19] .  \nThe graph analogue of the permanent–determinant contrast is the contrast between the perfectmatching polynomial and Pfaffians. Let G be a graph with 2n vertices, and assign a variable xe to each edge e ∈ E (G) . The perfect-matching polynomial of G is defined by  \nPerfMatch(G) = X Y xe ,  \nM∈M(G)e∈M  \nwhere M (G) denotes the set of perfect matchings of G. Setting all variables xe = 1, one obtains the number of perfect matchings of G. Thus PerfMatch(G) is the unsigned sum of the monomials corresponding to perfect matchings, just as the permanent is the unsigned sum over permutation terms.  \nIn the Pfaffian expansion, an orientation of the graph assigns signs to perfect matchings. Let D be an orientation of G, that is, a choice of direction for each","cbCaivOfmWwLNOoS","https://ap.wps.com/l/cbCaivOfmWwLNOoS","pdf",314804,2,1,15,"English","en",105,"# Introduction\n## Perfect matchings and computational complexity\n## Perfect-matching polynomial and Pfaffian expansions\n## Pfaffians, determinants, and Cayley’s identity","[{\"question\":\"What role does the FKT algorithm play in counting perfect matchings?\",\"answer\":\"The FKT algorithm counts perfect matchings in planar graphs in polynomial time using a single Pfaffian computation.\"},{\"question\":\"How do Galluccio–Loebl and Tesler generalize the Pfaffian method beyond planar graphs?\",\"answer\":\"For graphs embedded in an orientable surface of genus g, they show the perfect-matching polynomial can be written as a linear combination of at most 4g Pfaffians.\"},{\"question\":\"What does the paper prove about the Pfaffian number’s dependence on genus?\",\"answer\":\"It proves the exponential dependence is unavoidable: for every g ≥ 1 there exists a graph of orientable genus at most g whose perfect-matching polynomial requires at least (8/3)g Pfaffians in any linear representation.\"}]",1784175228,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"exponential-lower-bounds-for-the-pfaffian-number-of-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/exponential-lower-bounds-for-the-pfaffian-number-of-graphs/81658/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What role does the FKT algorithm play in counting perfect matchings?","Question",{"text":75,"@type":76},"The FKT algorithm counts perfect matchings in planar graphs in polynomial time using a single Pfaffian computation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do Galluccio–Loebl and Tesler generalize the Pfaffian method beyond planar graphs?",{"text":80,"@type":76},"For graphs embedded in an orientable surface of genus g, they show the perfect-matching polynomial can be written as a linear combination of at most 4g Pfaffians.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the paper prove about the Pfaffian number’s dependence on genus?",{"text":84,"@type":76},"It proves the exponential dependence is unavoidable: for every g ≥ 1 there exists a graph of orientable genus at most g whose perfect-matching polynomial requires at least (8/3)g Pfaffians in any linear 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