In the random subgraph model we consider random subgraphs G(t) of a graph G obtained as follows: for each edge in G we independently decide to retain the edge with probability t and discard the edge with probability 1 − t, for some 0 ≤ t ≤ 1. A special case of this model is the Erdős-Rényi random graph model, where the host graph is the complete graph Kn. In this paper we analyze the rigidity properties of random subgraphs and give new upper bounds on the threshold t0 for which Gt is a.a.s. rigid or globally rigid when t ≥ t0. By specializing our results to complete host graphs we obtain, among others, that an Erdős-Rényi random graph is a.a.s. globally rigid in Rd if t ≥ Cd logn n for some constant Cd. We also consider random subframeworks of (bar-and-joint) frameworks, which are geometric realizations of our graphs. Our bounds for the rigidity threshold of random subgraphs are in terms of the smallest non-zero eigenvalue of the stiffness matrix of the framework, which is the Gramian of its normalized rigidity matrix. Motivated by this connection, we introduce the concept of ddimensional algebraic connectivity of graphs and provide upper or lower bounds for this value of several fundamental graph classes. The case d = 1 corresponds to the well-known algebraic connectivity, that is, the second smallest Laplacian eigenvalue of the graph. We also consider the rigidity threshold in random molecular graphs, also called bond-bending networks, which are used in the study of rigidity properties of molecules. In this model we are concerned with the rigidity of the square graph of some graph G. We give an upper bound for the rigidity threshold of the square of random subgraphs in terms of the algebraic connectivity of the host graph. This enables us to derive an upper bound for the rigidity threshold for sparse host graphs. Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary. e-mail: jordan@cs.elte.hu Department of Mathematical Informatics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan. email: tanigawa@mist.i.u-tokyo.ac.jp
{"title":"Rigidity of Random Subgraphs and Eigenvalues of Stiffness Matrices","authors":"T. Jordán, Shin-ichi Tanigawa","doi":"10.1137/20m1349849","DOIUrl":"https://doi.org/10.1137/20m1349849","url":null,"abstract":"In the random subgraph model we consider random subgraphs G(t) of a graph G obtained as follows: for each edge in G we independently decide to retain the edge with probability t and discard the edge with probability 1 − t, for some 0 ≤ t ≤ 1. A special case of this model is the Erdős-Rényi random graph model, where the host graph is the complete graph Kn. In this paper we analyze the rigidity properties of random subgraphs and give new upper bounds on the threshold t0 for which Gt is a.a.s. rigid or globally rigid when t ≥ t0. By specializing our results to complete host graphs we obtain, among others, that an Erdős-Rényi random graph is a.a.s. globally rigid in Rd if t ≥ Cd logn n for some constant Cd. We also consider random subframeworks of (bar-and-joint) frameworks, which are geometric realizations of our graphs. Our bounds for the rigidity threshold of random subgraphs are in terms of the smallest non-zero eigenvalue of the stiffness matrix of the framework, which is the Gramian of its normalized rigidity matrix. Motivated by this connection, we introduce the concept of ddimensional algebraic connectivity of graphs and provide upper or lower bounds for this value of several fundamental graph classes. The case d = 1 corresponds to the well-known algebraic connectivity, that is, the second smallest Laplacian eigenvalue of the graph. We also consider the rigidity threshold in random molecular graphs, also called bond-bending networks, which are used in the study of rigidity properties of molecules. In this model we are concerned with the rigidity of the square graph of some graph G. We give an upper bound for the rigidity threshold of the square of random subgraphs in terms of the algebraic connectivity of the host graph. This enables us to derive an upper bound for the rigidity threshold for sparse host graphs. Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary. e-mail: jordan@cs.elte.hu Department of Mathematical Informatics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan. email: tanigawa@mist.i.u-tokyo.ac.jp","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"12 1","pages":"2367-2392"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81960974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G = (V,E) be a graph, and T ⊆ V a nonempty subset of even cardinality. The famous theorem of Edmonds and Johnson on the T -join polyhedron implies that the minimum cardinality of a T -cut is equal to the maximum value of a fractional packing of T -joins. In this paper, we prove that the fractions assigned may be picked as dyadic rationals, i.e. of the form a 2k for some integers a, k ≥ 0.
{"title":"On Dyadic Fractional Packings of $T$-Joins","authors":"Ahmad Abdi, G. Cornuéjols, Zuzanna Palion","doi":"10.1137/21m1445260","DOIUrl":"https://doi.org/10.1137/21m1445260","url":null,"abstract":"Let G = (V,E) be a graph, and T ⊆ V a nonempty subset of even cardinality. The famous theorem of Edmonds and Johnson on the T -join polyhedron implies that the minimum cardinality of a T -cut is equal to the maximum value of a fractional packing of T -joins. In this paper, we prove that the fractions assigned may be picked as dyadic rationals, i.e. of the form a 2k for some integers a, k ≥ 0.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"20 1","pages":"2445-2451"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86077148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Extremal Function for Excluding Geometry Minors over Prime Fields","authors":"P. Nelson, Zachary Walsh","doi":"10.1137/21m1410750","DOIUrl":"https://doi.org/10.1137/21m1410750","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"52 1","pages":"2100-2120"},"PeriodicalIF":0.0,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89866740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The class of cographs or complement-reducible graphs is the class of graphs that can be generated from K 1 using the operations of disjoint union and complementation. By analogy, this paper intro-duces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.
{"title":"The Smallest Classes of Binary and Ternary Matroids Closed under Direct Sums and Complements","authors":"J. Oxley, Jagdeep Singh","doi":"10.1137/21m1453852","DOIUrl":"https://doi.org/10.1137/21m1453852","url":null,"abstract":". The class of cographs or complement-reducible graphs is the class of graphs that can be generated from K 1 using the operations of disjoint union and complementation. By analogy, this paper intro-duces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"6 1","pages":"2051-2072"},"PeriodicalIF":0.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90333453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.
{"title":"The Flow Index of Regular Class I Graphs","authors":"Jiaao Li, Xueliang Li, Meiling Wang","doi":"10.1137/21m1393169","DOIUrl":"https://doi.org/10.1137/21m1393169","url":null,"abstract":"For integers k and d with k ≥ 2 d > 0, a circular k/d -flow of a graph G is an orientation together with a mapping from E ( G ) to {± d, ± ( d + 1) , . . . , ± ( k − d ) } such that, for each vertex of G , the sum of images on outgoing edges is equal to the sum of images on incoming edges. Related to the Four Color Problem, a classical result of Tutte shows that a cubic graph admits a circular 4 / 1-flow if and only if it is Class I (i.e., 3-edge-colorable). Tutte’s 3-flow conjecture implies that every 5-regular Class I graph admits a nowhere-zero 3-flow (equivalently, a circular 6 / 2-flow) as a special case. Steffen in 2015 conjectured that every (2 t + 1)-regular Class I graph admits a circular (2 t + 2) /t -flow. He also proposed a more general conjecture that every (2 t + 1)-odd-edge-connected (2 t + 1)-regular graph admits a circular (2 t + 2) /t -flow for any integer t ≥ 2, which includes the Circular Flow Conjecture of Jaeger(1981) stating that every 2 t -edge-connected graph admits a circular (2 t + 2) /t -flow for any even t ≥ 2. Jaeger’s conjecture was disproved in 2018 for all even t ≥ 6, and based on these results, Mattiolo and Steffen recently constructed counterexamples to Steffen’s conjecture for Class I graphs when t = 4 k + 2 for any integer k ≥ 1. -edge-connected (2 t +1)-regular Class I graphs without circular (2 t +2) /t -flows for any integer t ∈ { 6 , 8 , 10 } or t ≥ 12. Our result provides more general counterexamples to Steffen’s two conjectures for both even and odd t , and simultaneously generalizes the counterexamples of Jaeger’s Circular Flow Conjecture to regular Class I graphs.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"55 1","pages":"1991-2003"},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83873134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-24DOI: 10.48550/arXiv.2208.11573
Andr'e E. K'ezdy, JenHo Lehel
Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer'edi and Petruska proved $nleq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n leq {m+2 choose 2}$. This problem is known to be equivalent to determining the maximum order of a $tau$-critical $3$-uniform hypergraph with transversal number $m$ (details may also be found in a companion paper: arXiv:2204.02859). The best known bound, $nleq frac{3}{4}m^2+m+1$, was obtained by Tuza using the machinery of $tau$-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemer'edi and Petruska with the skew version of Bollob'as's theorem on set pair systems. The new approach improves the bound to $nleq {m+2 choose 2} + O(m^{{5}/{3}})$, resolving the conjecture asymptotically.
{"title":"An asymptotic resolution of a conjecture of Szemerédi and Petruska","authors":"Andr'e E. K'ezdy, JenHo Lehel","doi":"10.48550/arXiv.2208.11573","DOIUrl":"https://doi.org/10.48550/arXiv.2208.11573","url":null,"abstract":"Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer'edi and Petruska proved $nleq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n leq {m+2 choose 2}$. This problem is known to be equivalent to determining the maximum order of a $tau$-critical $3$-uniform hypergraph with transversal number $m$ (details may also be found in a companion paper: arXiv:2204.02859). The best known bound, $nleq frac{3}{4}m^2+m+1$, was obtained by Tuza using the machinery of $tau$-critical hypergraphs. Here we propose an alternative approach, a combination of the iterative decomposition process introduced by Szemer'edi and Petruska with the skew version of Bollob'as's theorem on set pair systems. The new approach improves the bound to $nleq {m+2 choose 2} + O(m^{{5}/{3}})$, resolving the conjecture asymptotically.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"24 1","pages":"113469"},"PeriodicalIF":0.0,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81237214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Small Maximal Sidon Set in ${mathbb{Z}}_2^n$","authors":"Maximus Redman, L. Rose, Raphael Walker","doi":"10.1137/21m1454663","DOIUrl":"https://doi.org/10.1137/21m1454663","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"686 1","pages":"1861-1867"},"PeriodicalIF":0.0,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76278722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Ideal Membership Problem over 3-Element CSPs with Dual Discriminator Polymorphism","authors":"Arpitha P. Bharathi, M. Mastrolilli","doi":"10.1137/21m1397131","DOIUrl":"https://doi.org/10.1137/21m1397131","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"55 1","pages":"1800-1822"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90624331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-02DOI: 10.48550/arXiv.2208.01470
Junxue Zhang
For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a classical extremal problem in graph theory. In 1962, ErdH{o}s determined {ex}$(K_n,kK_3)$, which generalized Mantel's Theorem. On the other hand, in 1974, {Bollob'{a}s}, ErdH{o}s, and Straus determined {ex}$(K_{n_1,n_2,dots,n_r},K_t)$, which extended Tur'{a}n's Theorem to complete multipartite graphs. { In this paper,} we determine {ex}$(K_{n_1,n_2,dots,n_r},kK_3)$ for $rge 4$ and $10k-4le n_1+4kle n_2le n_3le cdots le n_r$.
{"title":"Extremal numbers of disjoint triangles in r-partite graphs","authors":"Junxue Zhang","doi":"10.48550/arXiv.2208.01470","DOIUrl":"https://doi.org/10.48550/arXiv.2208.01470","url":null,"abstract":"For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a classical extremal problem in graph theory. In 1962, ErdH{o}s determined {ex}$(K_n,kK_3)$, which generalized Mantel's Theorem. On the other hand, in 1974, {Bollob'{a}s}, ErdH{o}s, and Straus determined {ex}$(K_{n_1,n_2,dots,n_r},K_t)$, which extended Tur'{a}n's Theorem to complete multipartite graphs. { In this paper,} we determine {ex}$(K_{n_1,n_2,dots,n_r},kK_3)$ for $rge 4$ and $10k-4le n_1+4kle n_2le n_3le cdots le n_r$.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"130 1","pages":"113530"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76408892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-01DOI: 10.48550/arXiv.2208.01001
N. Apollonio, Lorenzo Balzotti
Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.
{"title":"Two New Characterizations of Path Graphs","authors":"N. Apollonio, Lorenzo Balzotti","doi":"10.48550/arXiv.2208.01001","DOIUrl":"https://doi.org/10.48550/arXiv.2208.01001","url":null,"abstract":"Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"16 1","pages":"113596"},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89433486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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