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Anti-Ramsey Number of Edge-Disjoint Rainbow Spanning Trees
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-11-12 , DOI: 10.1137/19m1299876 Linyuan Lu , Zhiyu Wang
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-11-12 , DOI: 10.1137/19m1299876 Linyuan Lu , Zhiyu Wang
SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2346-2362, January 2020.
An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 82 (2016), pp. 75--89] conjectured that for any fixed $t$, $r(n,t)=(\begin{smallmatrix}{n-2}\\{2}\end{smallmatrix})+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$.
中文翻译:
边缘不相交的彩虹生成树的反Ramsey数
SIAM离散数学杂志,第34卷,第4期,第2346-2362页,2020年1月。
如果$ G $的每条边接收不同的颜色,则边色图$ G $被称为彩虹。$ t $边缘不相交的彩虹生成树的反Ramsey数,用$ r(n,t)$表示,定义为不包含$ t $边缘的$ K_n $边缘着色中的最大颜色数-不连贯的彩虹横跨树木。Jahanbekam和West [J. 图论,82(2016),第75--89页]推测对于任何固定的$ t $,$ r(n,t)=(\ begin {smallmatrix} {n-2} \\ {2} \ end {smallmatrix})+ t $只要$ n \ geq 2t + 2 \ geq 6 $。在本文中,我们证明了这一猜想。当$ n = 2t + 1 $时,我们也确定$ r(n,t)$。连同先前的结果,得出所有值$ n $和$ t $的$ t $边缘不相交彩虹生成树的反Ramsey数。
更新日期:2020-11-13
An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 82 (2016), pp. 75--89] conjectured that for any fixed $t$, $r(n,t)=(\begin{smallmatrix}{n-2}\\{2}\end{smallmatrix})+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$.
中文翻译:
边缘不相交的彩虹生成树的反Ramsey数
SIAM离散数学杂志,第34卷,第4期,第2346-2362页,2020年1月。
如果$ G $的每条边接收不同的颜色,则边色图$ G $被称为彩虹。$ t $边缘不相交的彩虹生成树的反Ramsey数,用$ r(n,t)$表示,定义为不包含$ t $边缘的$ K_n $边缘着色中的最大颜色数-不连贯的彩虹横跨树木。Jahanbekam和West [J. 图论,82(2016),第75--89页]推测对于任何固定的$ t $,$ r(n,t)=(\ begin {smallmatrix} {n-2} \\ {2} \ end {smallmatrix})+ t $只要$ n \ geq 2t + 2 \ geq 6 $。在本文中,我们证明了这一猜想。当$ n = 2t + 1 $时,我们也确定$ r(n,t)$。连同先前的结果,得出所有值$ n $和$ t $的$ t $边缘不相交彩虹生成树的反Ramsey数。