Approximation bounds on maximum edge 2-coloring of dense graphs

For a graph $G$ and integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {\em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $q\geq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,\ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + \frac {2} {\delta})$ for graphs with perfect matching and minimum degree $\delta$. For $\delta \ge 4$, this is better than the $\frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + \frac {1}{\delta - 1})$, which is better than $5/3$ for $\delta \ge 3$.Comment: 11pages, 3 figure

Stability of Myosin Subfragment-2 Modulates the Force Produced by Acto-Myosin Interaction of Striated Muscle

Myosin subfragment-2 (S2) is a coiled coil linker between myosin subfragment-1 and light meromyosin (LMM). This dissertation examines whether the myosin S2 coiled coil could regulate the amount of myosin S1 heads available to bind actin thin filaments by modulating the stability of its coiled coil. A stable myosin S2 coiled coil would have less active myosin S1 heads compared to a more flexible myosin S2 coiled coil, thus causing increased force production through acto-myosin interaction. The stability of the myosin S2 coiled coil was modulated by the binding of a natural myosin S2 binding protein, myosin binding protein C (MyBPC), and synthetic myosin S2 binding proteins, stabilizer and destabilizer peptide, to myosin S2. Competitive enzyme linked immunosorbent assay (cELISA) experiments revealed the cross specificity and high binding affinity of the synthetic peptides to the myosin S2 of human cardiac and rabbit skeletal origins. Gravitational force spectroscopy (GFS) was performed to test the stability of myosin S2 coiled coil in the presence of these myosin S2 binding proteins. GFS experiments demonstrated the stabilization of the myosin S2 coiled coil by the binding of MyBPC and stabilizer peptide to myosin S2, while the binding of destabilizer peptide to the same resulted in a flexible myosin S2 coiled coil. The binding of MyBPC and stabilizer peptide respectively, resulted in 3.35 and 1.5 times increase in force required to uncoil the myosin S2, while the binding of destabilizer peptide resulted in 1.6 times decrease in force required to uncoil the myosin S2. The myofibrillar contractility assay was performed to test the effect of synthetic myosin S2 binding proteins on the sarcomere shortening in myofibrils. The stabilizer peptide resulted in decreased sarcomere shortening of myofibrils as a result of decreased acto-myosin interaction, on the other hand, the binding of destabilizer peptide caused an increase in sarcomere shortening. The in vitro motility assay was performed to test the effect of altered stability of myosin S2 by binding of these myosin S2 binding proteins on the motility of actin filaments sliding over myosin. The motility of actin filaments was hindered by treating myosin thick filaments with whole length skeletal MyBPC or by treating heavy meromyosin with stabilizer peptide, while the motility of actin filaments was enhanced when heavy meromyosin was treated with destabilizer peptide. This study demonstrates that the myosin S2 coiled coil stability influences the force produced by acto-myosin interaction in striated skeletal muscle. The myosin S2 coiled coil when stabilized by MyBPC and stabilizer peptide resulted in decreased force production by reduced acto-myosin interaction. While the binding of destabilizer resulted in a flexible myosin S2 coiled coil and increased force production by enhanced acto-myosin interaction. The potentially cooperative response of contractility to the instability of the S2 coiled coil promises that this biological mechanism may be the target of drugs to modulate muscle performance

On additive combinatorics of permutations of ℤ_n

Special issue PRIMA 201

On additive combinatorics of permutations of ℤ_n

Special issue PRIMA 2013Let ℤn denote the ring of integers modulo n. A permutation of ℤn is a sequence of n distinct elements of ℤn. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤn, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2k≤s(n)≤n!· 2-(n-1)/2((n-1)/2)! and 2(n-1)/2·(n-1 / 2)!≤t(n)≤ 2k·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function

On Additive Combinatorics of Permutations of Z(n)

Let Z(n) denote the ring of integers modulo n. A permutation of Z(n) is a sequence of n distinct elements of Z(n). Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of Z(n), namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s (n) and t (n) respectively. The case when n is even is trivial in both the cases, with s (n) = 1 and t (n) = n!. For n odd, we prove (n phi(n))/2(k) <= s(n) <= n!.2(-)(n-1)/2/((n-1)/2)! and 2 (n-1)/2 . (n-1/2)! <= t (n) <= 2(k) . (n-1)!/phi(n), where k is the number of distinct prime divisors of n and phi is the Euler's totient function

बहुजन साहित्य की प्रस्तावना

International audienceThis book was originally published in Hindi with the title 'Bahujan Sahitya ki Prastavan" [Introduction to Bahujan Literature] in the year 2016. Its English translation was also published in the same year.This book presents a discussion on the concept of Bahujan literature in Hindi and Indian languages. On the one hand, this book monitors the changes taking place in Hindi literature, on the other hand, outlines the basic elements of the new debate that started in the field of literature in relation to the changing social and political scenarios after the implementation of the Mandal Commission. In today's Indian perspective, the Bahujan sections are – Other Backward Classes, Scheduled Tribes, Scheduled Castes, Denotified Nomadic Castes, Pasmanda Minorities and Women. The concept of Bahujan literature sees similarity in all these sufferings and finds the causes of their exploitation more or less similar. This book is as important for the researchers of literature as it is for the conscious people who aspire to understand the literary perspective and also for those who consider themselves a part of the Bahujan community. This book answers all the questions related to Bahujan discourse.यह पुस्तक मूल रूप से हिंदी में वर्ष 2016 में 'बहुजन साहित्य की प्रस्तावना' शीर्षक से प्रकाशित हुई थी। उसका यह अंग्रेजी अनुवाद भी उसी वर्ष प्रकाशित हुआ था।यह किताब हिन्दी और भारतीय भाषाओं में बहुजन साहित्य की अवधारणा पर विमर्श प्रस्तुत करती है। एक ओर यह किताब हिंदी साहित्य में हो रहे बदलावों पर नजर रखती है दूसरी ओर मंडल कमीशन के लागू होने के बाद बदल रहे समाज व राजनीतिक परिदृश्यों के सापेक्ष साहित्य के क्षेत्र में शुरू हुई नई बहस के बुनियादी तत्वों को रेखांकित भी करती है। आज के भारतीय परिप्रेक्ष्य में बहुजन तबके हैं- अन्य पिछड़ा वर्ग, अनुसूचित जनजातियां, अनुसूचित जातियां, विमुक्त घूमंतू जातियां, पसमांदा अल्पसंख्यक व स्त्रियां । बहुजन साहित्य की अवधारणा इन सभी की पीड़ाओं में समानता देखती है तथा इनके शोषण के कारणों को कमोबेश समान पाती है। यह किताब साहित्य के शोधार्थियों के लिए जितना महत्वपूर्ण है, उतना ही साहित्यिक नजरिए को समझने की आकंक्षा रखने वाले जागरूक लोगों के लिए भी और उनके लिए भी जो स्वयं को बहुजन का हिस्सा मानते हैं। फिर चाहे वे सामाजिक कार्यकर्ता हों या फिर राजनेता। बहुजन विमर्श से जुड‍़े सभी सवालों का यह किताब उत्तर देती है

बहुजन साहित्य की प्रस्तावना (हिंदी)

International audienceThis book presents a discussion on the concept of Bahujan literature in Hindi and Indian languages. On the one hand, this book monitors the changes taking place in Hindi literature, on the other hand, outlines the basic elements of the new debate that started in the field of literature in relation to the changing social and political scenarios after the implementation of the Mandal Commission. In today's Indian perspective, the Bahujan sections are – Other Backward Classes, Scheduled Tribes, Scheduled Castes, Denotified Nomadic Castes, Pasmanda Minorities and Women. The concept of Bahujan literature sees similarity in all these sufferings and finds the causes of their exploitation more or less similar. This book is as important for the researchers of literature as it is for the conscious people who aspire to understand the literary perspective and also for those who consider themselves a part of the Bahujan community. This book answers all the questions related to Bahujan discourse.his book was originally published in Hindi in the year 2016. Its English translation was also published in the same year, with the title "The Case For Bahujan Literature".The English translation of this book can be found on HAL here: https://hal.science/hal-03752562/यह किताब हिन्दी और भारतीय भाषाओं में बहुजन साहित्य की अवधारणा पर विमर्श प्रस्तुत करती है। एक ओर यह किताब हिंदी साहित्य में हो रहे बदलावों पर नजर रखती है दूसरी ओर मंडल कमीशन के लागू होने के बाद बदल रहे समाज व राजनीतिक परिदृश्यों के सापेक्ष साहित्य के क्षेत्र में शुरू हुई नई बहस के बुनियादी तत्वों को रेखांकित भी करती है। आज के भारतीय परिप्रेक्ष्य में बहुजन तबके हैं- अन्य पिछड़ा वर्ग, अनुसूचित जनजातियां, अनुसूचित जातियां, विमुक्त घूमंतू जातियां, पसमांदा अल्पसंख्यक व स्त्रियां । बहुजन साहित्य की अवधारणा इन सभी की पीड़ाओं में समानता देखती है तथा इनके शोषण के कारणों को कमोबेश समान पाती है। यह किताब साहित्य के शोधार्थियों के लिए जितना महत्वपूर्ण है, उतना ही साहित्यिक नजरिए को समझने की आकंक्षा रखने वाले जागरूक लोगों के लिए भी और उनके लिए भी जो स्वयं को बहुजन का हिस्सा मानते हैं। फिर चाहे वे सामाजिक कार्यकर्ता हों या फिर राजनेता। बहुजन विमर्श से जुड‍़े सभी सवालों का यह किताब उत्तर देती है।यह किताब वर्ष 2016 में हिंदी में प्रकाशित हुई थी। उसी साल इसका अंग्रेजी अनुवाद भी "The Case For Bahujan Literature" शीर्षक से प्रकाशित हुआ।इस किताब का अंग्रेजी अनुवाद HAL पर यहां देखा जा सकता है: https://hal.science/hal-03752562

महिषासुर: एक जननायक

International audienceThis is the English edition of the book 'Mahishasur: Ek Jannayak'. The book searches for a hero who has been ever-present in the memories, traditions, festivals, and celebrations of a majority of the people of India but who the dominant culture of the country has given the image of a villain. Recently, the search for this people's hero triggered a debate in Parliament. The book asks, "Why is there a celebration of death?" The Hindi version of this book can be found at HAL here: https://hal.science/hal-0374996