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[13] [14 <./figures/min-face-5-counterexample.png>] [15] Overfull \hbox (15.85718pt too wide) detected at line 1072 \OML/cmm/m/it/10 @F[] \OMS/cmsy/m/n/10 ^^R \OML/cmm/m/it/10 V\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ) \OMS/cmsy/m/n/10 [ \OML/cmm/m/it/10 V\OT 1/cmr/m/n/10 (\OML/cmm/m/it/10 K[]\OT1/cmr/m/n/10 ) [] [] [] [16] [17] Overfull \hbox (133.90562pt too wide) in paragraph at lines 1249--1257 \OT1/cmr/m/n/10 degree se-quence around $\OML/cmm/m/it/10 v$ \OT1/cmr/m/n/10 co n-tains at least one degree-$5$ en-try (see \OT1/cmtt/m/n/10 experiments/check[ ]v[]neighbour[]degrees.py\OT1/cmr/m/n/10 ). [] Overfull \hbox (86.1788pt too wide) in paragraph at lines 1269--1279 \OT1/cmr/bx/n/10 the chord-apex con-struc-tion). \OT1/cmr/m/n/10 Con-cretely (\ OT1/cmtt/m/n/10 experiments/check[]bad[]subcase[]deciding[]face.py\OT1/cmr/m/n/ 10 ), [] Overfull \hbox (2.93471pt too wide) in paragraph at lines 1294--1296 \OT1/cmr/bx/n/10 Lemma 5.21 \OT1/cmr/m/n/10 (Par-tial proof of Con-jec-ture 5.2 0[] for small un-cov-ered set)\OT1/cmr/bx/n/10 . \OT1/cmr/m/it/10 If $\OMS/cmsy /m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/10 ([]) \OMS/cmsy/m/n/10 n [] Overfull \hbox (4.00797pt too wide) in paragraph at lines 1312--1320 \OT1/cmtt/m/n/10 experiments/check[]S[]adjacency.py\OT1/cmr/m/n/10 , \OT1/cmtt/ m/n/10 experiments/check[]S[]face[]structure.py\OT1/cmr/m/n/10 ) [] [18] Overfull \hbox (12.19276pt too wide) in paragraph at lines 1324--1328 \OML/cmm/m/it/10 K[]; K[]$ \OT1/cmr/m/n/10 are the ``other'' Kempe cy-cles in t he $\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 c; c[]\OMS/cmsy/m/n/10 g$\OT1/cmr/m/n/10 - and $\OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 c; c[]\OMS/cmsy/m/n/10 g$\OT1/cmr/m/ n/10 -decompositions. [] Overfull \hbox (19.83406pt too wide) in paragraph at lines 1366--1372 \OT1/cmr/m/n/10 pentagon fall-back (Con-jec-ture 5.20[]) holds on $1[]314\OML/c mm/m/it/10 =\OT1/cmr/m/n/10 1[]314 = 100%$ of chord-apex+Kempe [] Overfull \hbox (158.34045pt too wide) in paragraph at lines 1366--1372 \OT1/cmr/m/n/10 colour-ings on which the par-tial struc-tural proof leaves a ga p (\OT1/cmtt/m/n/10 experiments/check[]gprime[]pentagon[]always[]works.py\OT1/c mr/m/n/10 ). [] [19] Overfull \hbox (68.54941pt too wide) in paragraph at lines 1452--1454 \OT1/cmr/m/n/10 (Every length ap-pear-ing is $\OMS/cmsy/m/n/10 6^^Q \OT1/cmr/m/ n/10 0([]3)$, as re-quired.) See \OT1/cmtt/m/n/10 experiments/check[]deciding[] face.py\OT1/cmr/m/n/10 . [] Underfull \hbox (badness 1052) in paragraph at lines 1469--1480 \OT1/cmr/m/n/10 this claim com-pu-ta-tion-ally on all chord-apex+Kempe colour-i ngs of re-duced [] Underfull \hbox (badness 3713) in paragraph at lines 1469--1480 \OT1/cmr/m/n/10 du-als with $\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 V\OT1/cmr/m/n/1 0 (\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 j ^^T \OT1/cmr/m/n/10 21 $; see \OT1/cmtt/m/n/10 experiments/check[]heawood[]on[]kempe.py\OT1/cmr/m/n/10 , [] Underfull \hbox (badness 10000) in paragraph at lines 1469--1480 \OT1/cmtt/m/n/10 experiments/check[]constancy[]obstruction.py\OT1/cmr/m/n/10 , and [] Overfull \hbox (45.67143pt too wide) in paragraph at lines 1483--1498 [] [] [20] [21] Underfull \hbox (badness 1112) in paragraph at lines 1603--1610 \OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.27\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 The strength-ened con-jec-ture was tested on the same chord- [] Underfull \hbox (badness 3492) in paragraph at lines 1603--1610 \OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.25[]; for each colour-ing w e sought [] Underfull \hbox (badness 10000) in paragraph at lines 1603--1610 \OT1/cmr/m/n/10 any Conjecture-5.1[]-witness $(\OML/cmm/m/it/10 F; e[]; e[]\OT1 /cmr/m/n/10 )$ whose ac-com-pa-ny-ing $\OML/cmm/m/it/10 f[]$ \OT1/cmr/m/n/10 sa t-is-fies [] Underfull \vbox (badness 4279) has occurred while \output is active [] [22] [23] (./paper.aux) LaTeX Font Warning: Some font shapes were not available, defaults substituted. ) Here is how much of TeX's memory you used: 3174 strings out of 478268 45923 string characters out of 5846347 350679 words of memory out of 5000000 21190 multiletter control sequences out of 15000+600000 481842 words of font info for 75 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 69i,13n,76p,1047b,404s stack positions out of 10000i,1000n,20000p,200000b,200000s < /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx9.pfb> < /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb> Output written on paper.pdf (23 pages, 1329304 bytes). 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