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Counting Homomorphisms to K4-minor-free Graphs, modulo 2
[chapter]

2021
*
Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)
*

As special cases, our machinery also yields a proof of the conjecture for

doi:10.1137/1.9781611976465.137
fatcat:mdxd2toiofd4pmgy6dhibm7jwq
*graphs*with maximum degree at most 3, as well as a full classification for the problem of*counting*list*homomorphisms*,*modulo**2*... We study the problem of computing the parity of the number of*homomorphisms*from an input*graph*G*to*a fixed*graph*H. ... Furthermore we thank Holger Dell for pointing out the tight conditional lower bound for*counting*independent sets*modulo**2*in [5] . ...##
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Goursat's theorem and the Zassenhaus lemma

1958
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

It is now easily shown that if A and B are sets with a limit operation satisfying Kl, K3 and

doi:10.4153/cjm-1958-005-6
fatcat:s2e6uspy5jgpld4qm3uhzemztq
*K4*, then any one-*to*-one correspondence between A and B is continuous if it has a closed*graph*. ... (for example, 1, 20) do not*count*division among the operations; the first*to*do so was apparently Evans (7) . ...##
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ARITHMETIC OF POTTS MODEL HYPERSURFACES

2013
*
International Journal of Geometric Methods in Modern Physics (IJGMMP)
*

We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the

doi:10.1142/s0219887813500059
fatcat:nwiidtv435hujalrq53b3y6fma
*graph*hypersurfaces of perturbative quantum field theory defined by the Kirchhoff*graph*... We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of*graphs*(Potts model partition function). ... In the case of the Ising model, where q =*2*, the*counting*function N p (Z*K4*,q=*2*) is the one computed in (5.11). ...##
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Arithmetic of Potts model hypersurfaces
[article]

2011
*
arXiv
*
pre-print

We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the

arXiv:1112.5667v1
fatcat:lnfqyyku6bhgjiomxntg74h3oi
*graph*hypersurfaces of perturbative quantum field theory defined by the Kirchhoff*graph*... We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of*graphs*(Potts model partition function). ... In the case of the Ising model, where q =*2*, the*counting*function N p (Z*K4*,q=*2*) is the one computed in (5.11). ...##
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Ideals of Graph Homomorphisms
[article]

2011
*
arXiv
*
pre-print

Keeping the categorical structure of

arXiv:1002.4679v5
fatcat:snj2piwshzadbnbmmxabp6erki
*graphs*in mind we give previous results a more functorial context and generalize them by introducing the ideals of*graph**homomorphisms*. ... In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from*graphs*. ... If k ≥*2*then µ(I C 2k →⫰ ) = k. Proof. Consider the cycle C 2k with vertices 0, 1, . . . 2k − 1 and edges {v, v + 1}*counting**modulo*2k. ...##
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Homomorphism preservation theorems

2008
*
Journal of the ACM
*

For a finite structure B, let CSP(B) denote the class of finite structures A that have a

doi:10.1145/1379759.1379763
fatcat:p3h7pkpk7ngvbmjx7wsrx6s3lq
*homomorphism**to*B. ... .: we show that a first-order formula is preserved under*homomorphisms*on all structures if and only if it is equivalent*to*an existential-positive formula of equal quantifier-rank. equivalent in the finite ... . ,*2*n } and relations E A = {(i, j) : i−j ≡ 1 or −1*modulo**2*n +1} and P A = {0,*2*, 4, . . . ,*2*n −*2*,*2*n }. ...##
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Topology of real algebraic curves near the non-singular tropical limit
[article]

2021
*
arXiv
*
pre-print

In that case, we

arXiv:2111.08607v1
fatcat:x2om2rebcfajxmndasj6bepx3q
*count*the number of even and odd ovals of those curves in terms of the dual subdivision, and construct some new counter-examples*to*Ragsdale's conjecture. ... Adding the*minor*M*to*the*minor*of D T given by the same rows and columns as M , we obtain the Laplacian mod*2*of the*graph*K m,n (respectively K l,m,n ) as a*minor*of Q Γ T , hence K m,n (respectively ...*to*the*minor*Q are all zero. ...##
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Turán H-densities for 3-graphs
[article]

2012
*
arXiv
*
pre-print

Given an $r$-

arXiv:1201.4326v1
fatcat:id5mlvwvynal3gd6e73v5n2rve
*graph*$H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$*to*be the maximum number of induced copies of $H$ in an $\mathcal{F}$-*free*... We also establish that \[\pi_{4.2}(\emptyset)=3/4,\] where 4.2 denotes the 3-*graph*on 4 vertices with exactly*2*edges. ... Acknowledgement We would like*to*thank Colin Reid for coming up with the idea that led*to*the lower bound construction in Theorem 18. ...##
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How is a graph like a manifold?
[article]

2002
*
arXiv
*
pre-print

Another example is the classical parallel redrawing problem, which turns out

arXiv:math/0206103v1
fatcat:qbxhid6w7nhlxkaj5bf5prmt4i
*to*be closely related*to*the problem of computing the second Betti number of a complex compact (^*)^n-manifold. ... In this article, we discuss some classical problems in combinatorics which can be solved by exploiting analogues between*graph*theory and the theory of manifolds. ... This shows the connection we defined above on the*graph**K4*. When we think of K 4 just as an abstract 4-regular*graph*we find that it has 10 different connections (up*to**graph*automorphism). ...##
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Carving-width and contraction trees for tensor networks
[article]

2019
*
arXiv
*
pre-print

We study the problem of finding contraction orderings on tensor networks for physical simulations using a syncretic abstract data type, the contraction-tree, and explain its connection

arXiv:1908.11034v1
fatcat:kgmfmsaf7navzox2kkrrrdxeay
*to*temporal and ... We have implemented the Ratcatcher of Seymour and Thomas for determining the carving-width of planar networks, in order*to*offer experimental evidence that this measure of spatial complexity makes a generally ... permission*to*redistribute Netcon; and Eduardo Mucciolo for sharing the graphics of figs. 6 and 7, as well as the term 'bottleneck.' ...##
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Around generalized external zomotopal algebras of graphs
[article]

2022
*
arXiv
*
pre-print

Following [NSh] we introduce and study a family of deformations of external zonotopal algebras associated

arXiv:2204.11331v1
fatcat:mskkbr3sbrgoni7f5lecuhxoiq
*to*a given undirected*graph*with possibly multiple edges, but no loops. ... We discuss different properties of the Hilbert series of these algebras and present the results of many calculations carried out in Macaulay*2*. Several conjectures are formulated. ... For G = K 4 , P A*K4*is a weighted projective line (topologically S*2*). ...##
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On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

2021

First, we introduce the study of quantum

doi:10.4230/lipics.icalp.2021.91
fatcat:hqmttj7zuzapxnkhvrkxgmziki
*graphs**to*the study of modular*counting**homomorphisms*. ... We study the problem of*counting*the number of*homomorphisms*from an input*graph*G*to*a fixed (quantum)*graph*̄{H} in any finite field of prime order ℤ_p. ... Mod Trees Cactus Square-*K4*-*minor*-pK3,3zteu, dominoqfree*free**free*Faben and Jerrum [21] 2Ĝ öbel et al. [26] 2ˆĜ öbel et al. [27] 2ˆF ocke et al. [23]*2ˆˆ*(ˆ)Ĝ öbel et al. ...##
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Weak containment of measure preserving group actions
[article]

2019
*
arXiv
*
pre-print

The case of the shift on [0, 1] Γ can be proved with

arXiv:1611.07921v5
fatcat:fa3j6jziefh7tjwjvfbtolftfm
*minor*modifications. ... The case of s Γ can be proved with*minor*modifications. Thus among the*free*actions there is a minimum, s Γ , and a maximum a ∞,Γ , in the sense of weak containment. ...*To*see that (iii) implies (ii), note that if n a n /U is not*free*and s Γ ⊑ n a n /U, then there is non-*free*a ∈ A(Γ, X, µ) with s Γ ⊑ a ⊑ n a n /U, contradicting Theorem 3.4. ...##
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The diamond lemma for ring theory

1978
*
Advances in Mathematics
*

A (fn = 8,) a (fn+l = b+,) (n

doi:10.1016/0001-8708(78)90010-5
fatcat:p2hmdqvysjabfncxxbxwqn3jky
*2*oh (24) this is congruent*modulo*I,,*to*Lx, [Y, 41 + [Y, 1% 41 + P, [%YlI, which is zero by the Jacobi identity in 3% So our ambiguities are resolvable relative*to*<. ... Then R is*free*as a right k-module OYZ the basis {zl ... Z, / n > 0, zi E*2*(x,), X, .*. X, E <X)i,}.1 1*to*x. ...##
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A SURVEY OF PROGRESS IN GRAPH THEORY IN THE SOVIET UNION

1970
*
Annals of the New York Academy of Sciences
*

The purpose of this report is

doi:10.1111/j.1749-6632.1970.tb56496.x
fatcat:cbb5x22w2farrnw5sijyovn66i
*to*draw attention*to*this collection of results, which are not well known in the West, and*to*summarize the significant contributions. ... Particular emphasis is placed upon those results that fill gaps o augment the body of knowledge about*graph*theory as familiar*to*non-Soviet specialists. ... These results are extended*to*what the author calls*2*-dimonbional symmetric functions, relating them by an isomorphism*to*the algebra of*graphs*(*modulo*q + 1). ...
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