A note on packing of graphic -tuples

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• 作者：Jian-Hua Yin ^{yinjh@ustc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Degree sequence ; Graphic nn-tuple ; Packing of graphic nn-tuples
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：132-137
• 全文大小：365 K

文摘

A nonnegative integer thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-tuple thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si3.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=169be69b88a607076fed42df35a72279" title="Click to view the MathML source">(d₁,…,d_n)thContainer hidden">thCode">th altimg="si3.gif" overflow="scroll">(d1,…,dn)th> (not necessarily monotone) is graphic   if there is a simple graph thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si4.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=1583f0b5a9fda2c219afb3d2f8ae514c" title="Click to view the MathML source">GthContainer hidden">thCode">th altimg="si4.gif" overflow="scroll">Gth> with the vertex set thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si5.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=c37d09af004c423ee77dd89c51a7b0ca" title="Click to view the MathML source">{v₁,…,v_n}thContainer hidden">thCode">th altimg="si5.gif" overflow="scroll">{v1,…,vn}th> in which the degree of thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si6.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=d32dbcd980208663b97310fd3d7b1161" title="Click to view the MathML source">v_ithContainer hidden">thCode">th altimg="si6.gif" overflow="scroll">vith> is thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si7.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=6ee54ba88dc1c9cd5a8373484343cb4b" title="Click to view the MathML source">d_ithContainer hidden">thCode">th altimg="si7.gif" overflow="scroll">dith>. Graphic thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-tuples thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si9.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=c69982fb2db9dc477685791ed966b618">th="97" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si9.gif">thContainer hidden">thCode">th altimg="si9.gif" overflow="scroll">(d1(1),…,dn(1))th> and thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si10.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=18a9d8d0debb035974f03cbbdfbabebc">th="97" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si10.gif">thContainer hidden">thCode">th altimg="si10.gif" overflow="scroll">(d1(2),…,dn(2))th>pack   if there are edge-disjoint thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-vertex graphs thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si12.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=3ab6fa4fe4729ed26a3143a03aeff3ba" title="Click to view the MathML source">G₁thContainer hidden">thCode">th altimg="si12.gif" overflow="scroll">G1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si13.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=707f5b6a0bf4a0e5647023d5c236a254" title="Click to view the MathML source">G₂thContainer hidden">thCode">th altimg="si13.gif" overflow="scroll">G2th> with the same vertex set thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si5.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=c37d09af004c423ee77dd89c51a7b0ca" title="Click to view the MathML source">{v₁,…,v_n}thContainer hidden">thCode">th altimg="si5.gif" overflow="scroll">{v1,…,vn}th> such that thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si15.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67bb189e75f1786727c2372b6635e8af">th="91" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si15.gif">thContainer hidden">thCode">th altimg="si15.gif" overflow="scroll">dG1(vi)=di(1)th> and thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si16.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=5ab6e42d3451796444a1f0a1f4ea021c">th="91" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si16.gif">thContainer hidden">thCode">th altimg="si16.gif" overflow="scroll">dG2(vi)=di(2)th> for all thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si17.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=aec97ec3faf11c2407984c66bd834188" title="Click to view the MathML source">ithContainer hidden">thCode">th altimg="si17.gif" overflow="scroll">ith>. Let thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si18.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=08b51541318514a5fbd82fc606ec0442" title="Click to view the MathML source">Δ(π)thContainer hidden">thCode">th altimg="si18.gif" overflow="scroll">Δ(π)th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si19.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=684b19a350b3e11e7ce8d5d6ac97ede7" title="Click to view the MathML source">δ(π)thContainer hidden">thCode">th altimg="si19.gif" overflow="scroll">δ(π)th> denote the largest and smallest entries in thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-tuple thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si21.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=69a3731ed140c507c141b6d74d914b28" title="Click to view the MathML source">πthContainer hidden">thCode">th altimg="si21.gif" overflow="scroll">πth> respectively. For graphic thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-tuples thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th>, Busch et al. (2012) proved that if thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si25.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=1a4a2ec47bb98790d1281b8a6503807e">th="340" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si25.gif">thContainer hidden">thCode">th altimg="si25.gif" overflow="scroll">Δ(π1+π2)≤2δ(π1+π2)n−(δ(π1+π2)−1)th> (strict inequality when thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si26.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=116860e8d016a08bb5f60cebc0980c2d" title="Click to view the MathML source">δ(π₁+π₂)=1thContainer hidden">thCode">th altimg="si26.gif" overflow="scroll">δ(π1+π2)=1th>), then thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> pack. As a more direct analogue to the Sauer–Spencer Theorem, Busch et al. conjectured that if thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si29.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=111d7c47f21f80684bdbbf26c0887edb" title="Click to view the MathML source">δ(π₁+π₂)≥1thContainer hidden">thCode">th altimg="si29.gif" overflow="scroll">δ(π1+π2)≥1th> and the product of corresponding entries in thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> is always less than thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si32.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=76af8ab216d5ac2144365e91a1c9d129">th="8" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si32.gif">thContainer hidden">thCode">th altimg="si32.gif" overflow="scroll">n2th>, then thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> pack. In this paper, we prove that if thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si35.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=a6697ce45abd572fe33f270d566b334b" title="Click to view the MathML source">δ(π₁)≥1thContainer hidden">thCode">th altimg="si35.gif" overflow="scroll">δ(π1)≥1th>, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> is an almost thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si37.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=8dd656882a49e9b1208cefe05fbe028d" title="Click to view the MathML source">kthContainer hidden">thCode">th altimg="si37.gif" overflow="scroll">kth>-regular graphic thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si1.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=67ba8d2d0dbf5d5a1a68797c0d0f469e" title="Click to view the MathML source">nthContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">nth>-tuple with thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si39.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=01df82fa5f04c4753f8ba034e6905efc" title="Click to view the MathML source">k≥1thContainer hidden">thCode">th altimg="si39.gif" overflow="scroll">k≥1th>, and the product of corresponding entries in thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> is always less than thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si32.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=76af8ab216d5ac2144365e91a1c9d129">th="8" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si32.gif">thContainer hidden">thCode">th altimg="si32.gif" overflow="scroll">n2th>, then thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> pack. This is an interesting variation of Kundu’s Theorem. Combining this result with a counterexample to the above conjecture of Busch et al., we present a slight modification of this conjecture as follows: if thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si35.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=a6697ce45abd572fe33f270d566b334b" title="Click to view the MathML source">δ(π₁)≥1thContainer hidden">thCode">th altimg="si35.gif" overflow="scroll">δ(π1)≥1th>, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si46.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=c89fc111d217b4158ebef8b0023a8fb5" title="Click to view the MathML source">δ(π₂)≥1thContainer hidden">thCode">th altimg="si46.gif" overflow="scroll">δ(π2)≥1th> and the product of corresponding entries in thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> is always less than thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si32.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=76af8ab216d5ac2144365e91a1c9d129">th="8" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X15002848-si32.gif">thContainer hidden">thCode">th altimg="si32.gif" overflow="scroll">n2th>, then thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si23.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=7339bd3e4c7440372fcb98b31f94f3f2" title="Click to view the MathML source">π₁thContainer hidden">thCode">th altimg="si23.gif" overflow="scroll">π1th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X15002848&_mathId=si24.gif&_user=111111111&_pii=S0012365X15002848&_rdoc=1&_issn=0012365X&md5=38bb9727bc9108d8a3fcd7e6b7d29b2e" title="Click to view the MathML source">π₂thContainer hidden">thCode">th altimg="si24.gif" overflow="scroll">π2th> pack.