A tight bound on the length of odd cycles in the incompatibility graph of a non-C1P matrix

AbstractIt is known that solar wind velocity fluctuates regularly with a period of about 1.3 years. This periodicity (and other signals with periods near to 1.1 and 0.9 years) has also been observed in biological data. The variation is a temporary feature, mostly being observed in the early 1990s. Here, the occurrence of these periodic signals in solar and geomagnetic activity between 1932 and 2005 has been investigated. The signal with 1.3 year period is present in geomagnetic activity only in a short interval after 1990 and to a lesser extent around 1942. At other times the signal is very weak or not present at all. Other periods are much lower amplitude and appear only sporadically throughout the time investigated. A connection between these periods and solar cycles (e.g. different even or odd cycles) has not been proven. It is possible that there is a long-term periodicity in the occurrence of the 1.3 year period but the time series data available is insufficient to confirm this. There are no such periodicities in solar activity. In order to gain a greater understanding of these periodic signals, we should search for their origin in interplanetary space.

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Between 2- and 3-Colorability

The Electronic Journal of Combinatorics ◽

10.37236/4673 ◽

2015 ◽

Vol 22 (1) ◽

Cited By ~ 1

Author(s):

Alan Frieze ◽

Wesley Pegden

Keyword(s):

Positive Integer ◽

Random Graphs ◽

The Other ◽

Chromatic Numbers ◽

Other Hand ◽

Odd Cycles

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$. These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.

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Tight Bound for the Number of Distinct Palindromes in a Tree

String Processing and Information Retrieval - Lecture Notes in Computer Science ◽

10.1007/978-3-319-23826-5_26 ◽

2015 ◽

pp. 270-276 ◽

Cited By ~ 2

Author(s):

Paweł Gawrychowski ◽

Tomasz Kociumaka ◽

Wojciech Rytter ◽

Tomasz Waleń

Keyword(s):

Tight Bound

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A note on compact and compact circular edge-colorings of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.431 ◽

2008 ◽

Vol Vol. 10 no. 3 (Graph and Algorithms) ◽

Author(s):

Dariusz Dereniowski ◽

Adam Nadolski

Keyword(s):

Approximation Algorithm ◽

Decision Problem ◽

Edge Coloring ◽

Exact Algorithm ◽

Weighted Graphs ◽

Edge Colorings ◽

Odd Cycle ◽

International Audience ◽

Np Complete ◽

Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

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On Length Independent Security Bounds for the PMAC Family

IACR Transactions on Symmetric Cryptology ◽

10.46586/tosc.v2021.i2.423-445 ◽

2021 ◽

pp. 423-445

Author(s):

Bishwajit Chakraborty ◽

Soumya Chattopadhyay ◽

Ashwin Jha ◽

Mridul Nandi

Keyword(s):

Block Cipher ◽

Block Size ◽

Gray Code ◽

Tight Bound ◽

Simple Modification ◽

Security Proof ◽

Tight Security ◽

Pseudorandom Function ◽

Exact Security ◽

Query Length

At FSE 2017, Gaži et al. demonstrated a pseudorandom function (PRF) distinguisher (Gaži et al., ToSC 2016(2)) on PMAC with Ω(lq2/2n) advantage, where q, l, and n, denote the number of queries, maximum permissible query length (in terms of n-bit blocks), and block size of the underlying block cipher. This, in combination with the upper bounds of Ο(lq2/2n) (Minematsu and Matsushima, FSE 2007) and Ο(qσ/2n) (Nandi and Mandal, J. Mathematical Cryptology 2008(2)), resolved the long-standing problem of exact security of PMAC. Gaži et al. also showed that the dependency on l can be dropped (i.e. O(q2/2n) bound up to l ≤ 2n/2) for a simplified version of PMAC, called sPMAC, by replacing the Gray code-based masking in PMAC with any 4-wise independent universal hash-based masking. Recently, Naito proposed another variant of PMAC with two powering-up maskings (Naito, ToSC 2019(2)) that achieves l-free bound of O(q2/2n), provided l ≤ 2n/2. In this work, we first identify a flaw in the analysis of Naito’s PMAC variant that invalidates the security proof. Apparently, the flaw is not easy to fix under the existing proof setup. We then formulate an equivalent problem which must be solved in order to achieve l-free security bounds for this variant. Second, we show that sPMAC achieves O(q2/2n) bound for a weaker notion of universality as compared to the earlier condition of 4-wise independence. Third, we analyze the security of PMAC1 (a popular variant of PMAC) with a simple modification in the linear combination of block cipher outputs. We show that this simple modification of PMAC1 has tight security O(q2/2n) provided l ≤ 2n/4. Even if l < 2n/4, we still achieve same tight bound as long as total number of blocks in all queries is less than 22n/3.

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Ramsey Numbers of Trees Versus Odd Cycles

The Electronic Journal of Combinatorics ◽

10.37236/5731 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 1

Author(s):

Matthew Brennan

Keyword(s):

Positive Integer ◽

Constant Factor ◽

Linear Functions ◽

Ramsey Numbers ◽

Odd Cycle ◽

Odd Cycles

Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.

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Vertex-Transitive Direct Products of Graphs

The Electronic Journal of Combinatorics ◽

10.37236/6999 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Richard H. Hammack ◽

Wilfried Imrich

Keyword(s):

Direct Product ◽

Direct Products ◽

Odd Cycle ◽

Vertex Transitive ◽

Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

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