On The Maximum Jump Number M(2k-1,k)


Abstract


If n and k (n\geq k) are large enough , it is   quite difficult to give the value of M(n,k). R.A. Brualdi and H.C. Jung gave   a table about the value of M(n,k) for 1\leq k \leq n\leq 10 . In this paper, we show that   4(k-1)-\lceil\sqrt{k-1}\rceil\leq M(2k-1,k)\leq 4k-7 holds for   k\geq 6. Hence, M(2k-1,k)=4k-7 holds for 6\leq k \leq 10,   which verifies that their conjecture   M(2k+1,k+1)=4k-\lceil\sqrt{k}\rceil holds for 5\leq k\leq 9,   and disprove their conjecture   M(n,k)<M(n+l_{1},k+l_{2}) for l_{1}= 1 , l_{2}= 1 .

DOI Code: 10.1285/i15900932v23n1p71

Keywords: (0,1)-matrices; Jump number; Stair number; Conjecture

Classification: 05B20; 15A36

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