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The Lower and Upper Bounds of Turán Number for Odd Wheels
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2021-03-10 , DOI: 10.1007/s00373-021-02290-0
Byeong Moon Kim , Byung Chul Song , Woonjae Hwang

The Turán number for a graph H, denoted by \(\text {ex}(n,H)\), is the maximum number of edges in any simple graph with n vertices which doesn’t contain H as a subgraph. In this paper we find the lower and upper bounds for \(\text { ex}(n,W_{2t+1})\). We show that if \(n\ge 4t\), then \(\text { ex}(n,W_{2t+1})\ge \left\lfloor \lfloor \frac{2n+t}{4}\rfloor (n+\frac{t-1}{2}-\lfloor \frac{2n+t}{4}\rfloor )\right\rfloor +1.\) We also show that for sufficiently large n and \(t\ge 5\), \(\text { ex}(n,W_{2t+1})\le \frac{ n^2 }{4}+{t-1\over 2}n\). Moreover we find the exact value of the Turán number for \(W_9\). That is, we show that for sufficiently large n, \(\text { ex}(n,W_9)= \lfloor \frac{n^2}{4}\rfloor +\lceil \frac{3}{4}n\rceil +1\).



中文翻译:

图奇数轮的Turán数的上下界

\(\ text {ex}(n,H)\)表示的图H的Turán数是具有n个顶点且不包含H作为子图的任何简单图的最大边数。在本文中,我们找到\(\ text {ex}(n,W_ {2t + 1})\)的上下限。我们证明如果\(n \ ge 4t \),则\(\ text {ex}(n,W_ {2t + 1})\ ge \ left \ lfloor \ lfloor \ frac {2n + t} {4} \ rfloor(n + \ frac {t-1} {2}-\ lfloor \ frac {2n + t} {4} \ rfloor)\ right \ rfloor +1。\)我们也证明了对于足够大的n\(t \ ge 5 \)\(\ text {ex}(n,W_ {2t + 1})\ le \ frac {n ^ 2} {4} + {t-1 \ over 2} n \)。此外,我们找到\(W_9 \)的图兰数的确切值。也就是说,我们表明对于足够大的n\(\ text {ex}(n,W_9)= \ lfloor \ frac {n ^ 2} {4} \ rfloor + \ lceil \ frac {3} {4} n \ rceil +1 \)

更新日期:2021-03-10
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