scholarly journals $b$-Invariant Edges in Essentially 4-Edge-Connected Near-Bipartite Cubic Bricks

10.37236/8945 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Fuliang Lu ◽  
Xing Feng ◽  
Yan Wang
Keyword(s):  
Lower Bound ◽  
Building Blocks ◽  
Petersen Graph ◽  
Cubic Graph ◽  
Bipartite Matching

A brick is a  non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is $b$-invariant if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is essentially 4-edge-connected if it does not contain nontrivial 3-cuts. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi  and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.

scholarly journals Multiple Petersen Subdivisions in Permutation Graphs

10.37236/2239 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Tomáš Kaiser ◽  
Jean-Sébastien Sereni ◽  
Zelealem B. Yilma
Keyword(s):  
Lower Bound ◽  
Constant Factor ◽  
Petersen Graph ◽  
Cubic Graph ◽  
Permutation Graph ◽  
Permutation Graphs

A permutation graph is a cubic graph admitting a 1-factor $M$ whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if $e$ is an edge of $M$ such that every 4-cycle containing an edge of $M$ contains $e$, then $e$ is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of $M$ is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor.

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

scholarly journals MIXED STATE ENTANGLEMENT MEASURES FOR INTERMEDIATE SEPARABILITY

2010 ◽  
Vol 08 (04) ◽  
pp. 677-685 ◽  
Author(s):  
TSUBASA ICHIKAWA ◽  
MARCUS HUBER ◽  
PHILIPP KRAMMER ◽  
BEATRIX C. HIESMAYR
Keyword(s):  
Lower Bound ◽  
Quantum State ◽  
Mixed State ◽  
Building Blocks ◽  
Mixed States ◽  
Entanglement Measures

To determine whether a given multipartite quantum state is separable with respect to some partition, we construct a family of entanglement measures {Rm(ρ)}. This is done utilizing generalized concurrences as building blocks which are defined by flipping of M constituents and indicate states that are separable with regard to bipartitions when vanishing. Furthermore, we provide an analytically computable lower bound for {Rm(ρ)} via a simple ordering relation of the convex roof extension. Using the derived lower bound, we illustrate the effect of the isotropic noise on a family of four-qubit mixed states for each intermediate separability.

scholarly journals On Essentially 4-Edge-Connected Cubic Bricks

10.37236/8594 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Nishad Kothari ◽  
Marcelo H. De Carvalho ◽  
Cláudio L. Lucchesi ◽  
Charles H. C. Little
Keyword(s):  
Petersen Graph ◽  
Triangular Prism ◽  
Cubic Graph

Lovász (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.) A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lovász which states that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge. A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected and if its only $3$-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact. We prove that if $G$ is any essentially $4$-edge-connected cubic brick then its edge-set may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii) quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$ (distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.

scholarly journals Some Results on the Structure of Multipoles in the Study of Snarks

10.37236/3629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. A. Fiol ◽  
J. Vilaltella
Keyword(s):  
Lower Bound ◽  
Edge Coloring ◽  
The State ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Order ◽  
Exact Number ◽  
Empty Subset ◽  
Free Edges

Multipoles are the pieces we obtain by cutting some edges of a cubic graph in one or more points. As a result of the cut, a multipole $M$ has vertices attached to a dangling edge with one free end, and isolated edges with two free ends. We refer to such free ends as semiedges, and to isolated edges as free edges. Every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, which satisfies the Parity Lemma. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable. Some results on the states and structure of the so-called color complete and color closed multipoles are presented. In particular, we give lower and upper linear bounds on the minimum order of a color complete multipole, and compute its exact number of states. Given two multipoles $M_1$ and $M_2$ with the same number of semiedges, we say that $M_1$ is reducible to $M_2$ if the state set of $M_2$ is a non-empty subset of the state set of $M_1$ and $M_2$ has less vertices than $M_1$. The function $v(m)$ is defined as the maximum number of vertices of an irreducible multipole with $m$ semiedges. The exact values of  $v(m)$ are only known for $m\le 5$. We prove that tree and cycle multipoles are irreducible and, as a byproduct, that $v(m)$ has a linear lower bound.

scholarly journals Constructions for Cubic Graphs with Large Girth

10.37236/1386 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Norman Biggs
Keyword(s):  
Lower Bound ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimal Graph ◽  
Minimum Value ◽  
Open Questions ◽  
G 12 ◽  
Large Girth ◽  
Selection Of ◽  
Finite Constant

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer $\mu_0(g)$, the smallest number of vertices for which a cubic graph with girth at least $g$ exists, and furthermore, the minimum value $\mu_0(g)$ is attained by a graph whose girth is exactly $g$. The values of $\mu_0(g)$ when $3 \le g \le 8$ have been known for over thirty years. For these values of $g$ each minimal graph is unique and, apart from the case $g=7$, a simple lower bound is attained. This paper is mainly concerned with what happens when $g \ge 9$, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if $g=12$. A number of techniques are described, with emphasis on the construction of families of graphs $\{ G_i\}$ for which the number of vertices $n_i$ and the girth $g_i$ are such that $n_i\le 2^{cg_i}$ for some finite constant $c$. The optimum value of $c$ is known to lie between $0.5$ and $0.75$. At the end of the paper there is a selection of open questions, several of them containing suggestions which might lead to improvements in the known results. There are also some historical notes on the current-best graphs for girth up to 36.

scholarly journals Small Snarks with Large Oddness

10.37236/3969 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Robert Lukoťka ◽  
Edita Máčajová ◽  
Ján Mazák ◽  
Martin Škoviera
Keyword(s):  
Infinite Family ◽  
Upper Bounds ◽  
Petersen Graph ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Minimum Number

We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\, \omega(G)$ vertices, and for each integer $k$ with $2\le k\le 6$ we construct an infinite family of cubic graphs with cyclic connectivity $k$ and small oddness ratio $|V(G)|/\omega(G)$. In particular, for cyclic connectivity $2$, $4$, $5$, and $6$ we improve the upper bounds on the oddness ratio of snarks to $7.5$, $13$, $25$, and $99$ from the known values $9$, $15$, $76$, and $118$, respectively. In addition, we construct a cyclically $4$-connected snark of girth $5$ with oddness $4$ on $44$ vertices, improving the best previous value of $46$. Corrigendum added March 19, 2018.

scholarly journals Improved Attacks on sLiSCP Permutation and Tight Bound of Limited Birthday Distinguishers

2020 ◽  
pp. 147-172
Author(s):  
Akinori Hosoyamada ◽  
María Naya-Plasencia ◽  
Yu Sasaki
Keyword(s):  
Lower Bound ◽  
Building Blocks ◽  
Upper Bounds ◽  
Middle Part ◽  
Tight Bound ◽  
External Conditions ◽  
Standardization Process ◽  
First Time

Limited birthday distinguishers (LBDs) are widely used tools for the cryptanalysis of cryptographic permutations. In this paper we propose LBDs on several variants of the sLiSCP permutation family that are building blocks of two round 2 candidates of the NIST lightweight standardization process: Spix and SpoC. We improve the number of steps with respect to the previously known best results, that used rebound attack. We improve the techniques used for solving the middle part, called inbound, and we relax the external conditions in order to extend the previous attacks. The lower bound of the complexity of LBDs has been proved only against functions. In this paper, we prove for the first time the bound against permutations, which shows that the known upper bounds are tight.

scholarly journals Lightweight Iterative MDS Matrices: How Small Can We Go?

2020 ◽  
pp. 147-170
Author(s):  
Shun Li ◽  
Siwei Sun ◽  
Danping Shi ◽  
Chaoyun Li ◽  
Lei Hu
Keyword(s):  
Lower Bound ◽  
Design Space ◽  
Low Cost ◽  
Optimal Solution ◽  
Building Blocks ◽  
Block Matrix ◽  
Diffusion Layers ◽  
Binary Matrices ◽  
Symmetric Key ◽  
Iterative Construction

As perfect building blocks for the diffusion layers of many symmetric-key primitives, the construction of MDS matrices with lightweight circuits has received much attention from the symmetric-key community. One promising way of realizing low-cost MDS matrices is based on the iterative construction: a low-cost matrix becomes MDS after rising it to a certain power. To be more specific, if At is MDS, then one can implement A instead of At to achieve the MDS property at the expense of an increased latency with t clock cycles. In this work, we identify the exact lower bound of the number of nonzero blocks for a 4 × 4 block matrix to be potentially iterative-MDS. Subsequently, we show that the theoretically lightest 4 × 4 iterative MDS block matrix (whose entries or blocks are 4 × 4 binary matrices) with minimal nonzero blocks costs at least 3 XOR gates, and a concrete example achieving the 3-XOR bound is provided. Moreover, we prove that there is no hope for previous constructions (GFS, LFS, DSI, and spares DSI) to beat this bound. Since the circuit latency is another important factor, we also consider the lower bound of the number of iterations for certain iterative MDS matrices. Guided by these bounds and based on the ideas employed to identify them, we explore the design space of lightweight iterative MDS matrices with other dimensions and report on improved results. Whenever we are unable to find better results, we try to determine the bound of the optimal solution. As a result, the optimality of some previous results is proved.

scholarly journals Lightweight Diffusion Layer: Importance of Toeplitz Matrices

2016 ◽  
pp. 95-113 ◽  
Author(s):  
Sumanta Sarkar ◽  
Habeeb Syed
Keyword(s):  
Lower Bound ◽  
Diffusion Layer ◽  
Hardware Implementation ◽  
Toeplitz Matrices ◽  
Building Blocks ◽  
Block Ciphers ◽  
Implementation Cost ◽  
Mds Matrix ◽  
Minimum Value ◽  
Diffusion Layers

MDS matrices are used as building blocks of diffusion layers in block ciphers, and XOR count is a metric that estimates the hardware implementation cost. In this paper we report the minimum value of XOR counts of 4 × 4 MDS matrices over F24 and F28 , respectively. We give theoretical constructions of Toeplitz MDS matrices and show that they achieve the minimum XOR count. We also prove that Toeplitz matrices cannot be both MDS and involutory. Further we give theoretical constructions of 4 × 4 involutory MDS matrices over F24 and F28 that have the best known XOR counts so far: for F24 our construction gives an involutory MDS matrix that actually improves the existing lower bound of XOR count, whereas for F28 , it meets the known lower bound.

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