Haj\'os conjectured that every graph containing no subdivision of the complete $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, maximum chromatic number such graphs $\Omega(s^2/\log s)$. We prove $O(s)$ colors are enough for a weakening this only requires monochromatic component to have bounded size (so-called clustered coloring). Our approach leads more res...

The performance of music usually involves a great deal of interpretation by the musician. In classical music, the final ritardando is a good example of the expressive aspect of music performance. Even though expressive timing data is expected to have a strong component that is determined by the piece itself, in this paper we investigate to what degree individual performance style has an effect ...

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln)  mk1 is odd grace...

Dynamics of eternal inflation on the landscape admits description in terms of the Martin-Siggia-Rose (MSR) effective field theory that is in one-to-one correspondence with vacuum dynamics equations. On those sectors of the landscape, where transport properties of the probability measure for eternal inflation are important, renormalization group fixed points of the MSR effective action determine...