Cycles through an arc in regular 3-partite tournaments

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• 作者：Qiaoping Guo ; ^{guoqp@sxu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Linan Cui ; Wei Meng
• 关键词：Multipartite tournament ; Regular multipartite tournament ; 3-partite tournament ; Cycle
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：17-20
• 全文大小：355 K

文摘

A 365X15002447&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=4f5710d5913cc4f2a1dd53855e377c9c" title="Click to view the MathML source">c$c$-partite tournament is an orientation of a complete 365X15002447&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=4f5710d5913cc4f2a1dd53855e377c9c" title="Click to view the MathML source">c$c$-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament 365X15002447&_mathId=si105.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=05bf2eb7445ca0a44f7a131fddca7789" title="Click to view the MathML source">D$D$ is contained in an 365X15002447&_mathId=si4.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=64ed41637ec06430c94fd3ad17db5f99" title="Click to view the MathML source">m$m$-, 365X15002447&_mathId=si5.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=51fca330e7cf1e06f2026b97db705863" title="Click to view the MathML source">(m+1)$(m+1)$- or 365X15002447&_mathId=si66.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=50bb370734183099c72d5d99a2c7eb02" title="Click to view the MathML source">(m+2)$(m+2)$-cycle for each 365X15002447&_mathId=si67.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=19d8d0f31755be0df2777a3785aa666f" title="Click to view the MathML source">m∈{3,4,…,|V(D)|−2}$m\in \{\mathrm{<font\; color="red">3</font>},4,\dots ,\left|V\left(D\right)\right|-2\}$, and he also proved this conjecture for 365X15002447&_mathId=si68.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=f5b923502859403156ca4901772ad22f" title="Click to view the MathML source">m=3,4,5$m=\mathrm{<font\; color="red">3</font>},4,5$. In 2012, Xu et al. proved that every arc of a regular 3-partite tournament is contained in a 5- or 6-cycle, and in the same paper, the authors also posed the following conjecture:

Conjecture 1. If 365X15002447&_mathId=si105.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=05bf2eb7445ca0a44f7a131fddca7789" title="Click to view the MathML source">D$D$ is an 365X15002447&_mathId=si106.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=fd36dabcddd71f43091e793e70654259" title="Click to view the MathML source">r$r$-regular 3-partite tournament with 365X15002447&_mathId=si11.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=270dac76728bc17a938d9ca485993e54" title="Click to view the MathML source">r≥2$r\ge 2$, then every arc of 365X15002447&_mathId=si105.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=05bf2eb7445ca0a44f7a131fddca7789" title="Click to view the MathML source">D$D$ is contained in a 3" class="mathmlsrc">365X15002447&_mathId=si13.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=44ab7ce162a774531e4778ac04c61a73" title="Click to view the MathML source">3k- or 365X15002447&_mathId=si14.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=3946a4f19476b767b0960437e9f4eec1" title="Click to view the MathML source">(3k+1)$(\mathrm{<font\; color="red">3</font>}k+1)$-cycle for 365X15002447&_mathId=si15.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=0cdf3dcae02bae262a39f247f0170c2e" title="Click to view the MathML source">k=1,2,…,r−1$k=1,2,\dots ,r-1$.

It is known that Conjecture 1 is true for 365X15002447&_mathId=si16.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=58a71e23ed631b2d5a52a1c61c8f3964" title="Click to view the MathML source">k=1$k=1$. In this paper, we prove Conjecture 1 for 365X15002447&_mathId=si17.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=6b2f76537b7495ae75a4e6100cc4f063" title="Click to view the MathML source">k=2$k=2$, which implies that Volkmann’s conjecture for 365X15002447&_mathId=si18.gif&_user=111111111&_pii=S0012365X15002447&_rdoc=1&_issn=0012365X&md5=7485a0076a4fd3f89a6e0de79499fd39" title="Click to view the MathML source">m=6$m=6$ is correct.