(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 10.1' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1064, 20] NotebookDataLength[ 2695084, 52757] NotebookOptionsPosition[ 2694404, 52717] NotebookOutlinePosition[ 2694975, 52738] CellTagsIndexPosition[ 2694932, 52735] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Testing 4-critical plane and projective plane\nmultiwheels using ", StyleBox["Mathematica", FontSlant->"Italic"], " \n" }], "Title", CellChangeTimes->{{3.6544093176711917`*^9, 3.6544093708944693`*^9}, { 3.65485630271259*^9, 3.6548563083347845`*^9}, 3.6548563403439045`*^9}], Cell["\<\ Dainis Zeps, IMCS, University of Latvia Abstract\ \>", "Subtitle", CellChangeTimes->{{3.654409379974913*^9, 3.6544094527102165`*^9}, { 3.654855043392466*^9, 3.6548550501297197`*^9}, 3.654855120123832*^9, { 3.6548551582938733`*^9, 3.6548551597476606`*^9}}], Cell[TextData[{ "In this article we explore 4-critical graphs using ", StyleBox["Mathematica", FontSlant->"Italic"], ". We generate graph patterns according [1, D. Zeps. On building 4-critical \ plane and projective plane multiwheels from odd wheels, arXiv:1202.4862v1]. \ Using the base graph, minimal planar multiwheel and in the same time minimal \ according projective pattern built multiwheel, we build minimal multiwheels \ according [1], We forward two conjectures according graphs augmented \ according considered patterns and their 4-criticallity, and argue them to be \ proved here if the paradigmatic examples of this article are accepted to be \ parts of proofs. 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Thus, the smallest built by us multiwheel is ", Cell[BoxData[ FormBox[ SubscriptBox["w", "111"], TraditionalForm]]], " built from three wheels ", Cell[BoxData[ FormBox[ SubscriptBox["W", "3"], TraditionalForm]]], ". Note, we design ordinary wheels with capital letters W, and multiwheels \ with small letters w. 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By hand is done what we see \ right and in two versions: planar and as Gr\[ODoubleDot]tsch series graph.\n\n\ Smallest of our multiwheels ", Cell[BoxData[ FormBox[ SubscriptBox["w", "111"], TraditionalForm]]], ". Central hub is vertex 1, section hubs are 2,4,7, and rim vertices are \ 5,6,7. This graph is ", Cell[BoxData[ FormBox[ SubscriptBox["g", "111"], TraditionalForm]]], " two and on projective plane it quadrangulates it. Central hub spikes unite \ central hub with section hubs, Section hub spikes unite section hubs with rim \ vertices, and rim spikes unite rim vertices between themselves. This we call \ the base graph, taking it as paradigmical, because all other multiwheels are \ built increaseing numbers s and ", Cell[BoxData[ FormBox[ SubscriptBox["k", "i"], TraditionalForm]]], "-s. The only variation from this is that sections may be implemented in \ four ways, see page 6. [1]. \nFurther in the article we test multiwheels that \ we build on being 4-critical. It must be said that we as if proof this \ criticality in the article [1], but we relay on article [2] that is not yet \ published and even ready. 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Let us \ take the base graph and replace each simple section with the base graph. We \ do in all possible ways for section orientation in different variations. \n\ Let us fix notation for base graph edges: the base has three types of edges: \ central spike or c-spike, section spike or s-spike and rim edge or r-edge, \ thus, c-spike, s-spike and r-edge. Central hub is vertex 1, section hubs are \ 2,3,4, and rim vertices are 5,6,7. 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But \ the proofs in [1] relay on the promise that the article [3] should provide \ the truth of the fact that graph can\[CloseCurlyQuote]t remain k-critical \ after vertex split. We here testet base graph\[CloseCurlyQuote]s main \ configurations on being 4-critical, and [1] shurely argues that simpply \ augmented configurations with section number augmentaion without changing \ their type can\[CloseCurlyQuote]t disturb graph being 4-critical. Thus, \ testing base graph configurations here we conclude the proofs of [1] without \ assumtion of [3]. Here even more, we test multiwheel configurations with \ sections being multiwheels, here, base graph. We must take into \ considerations that we can augment graph as planar of as Gr\[ODoubleDot]tsch \ graph because the base graph belongs to both clases, i.e., is the minimal \ graph in both series, and both series augmentations should equaly work in \ producing 4-critical graphs. But we found two exeptional graphs, situations. Section with s-spike rim \ vertex s-spike marking gave 3-chromatic graph that of course isn\ \[CloseCurlyQuote]t 4-critical. But in the same time it is easy to see that \ this graph, and all augments from it, doesn\[CloseCurlyQuote]t contain \ octahedral bracket as graph invariant: simply, octahedron is their minor. \ Alas, one more configuraion gives graphs with O as minor. This was r-edge \ r-edge case, which gave 4-critical graphs. But these facts allow to forward \ following conjectures: Conjecture 1: Multiwheels received in the way discribed in [1] and here are 4-critical, or \ 3-chromatic and with O as minor. Conjecture 2: [3] holds for 4-critical multiwheels built according patterns of [1] and what \ we find here. \ \>", "Text", CellChangeTimes->{{3.6548626038799753`*^9, 3.6548626506481695`*^9}, { 3.6548627102909355`*^9, 3.654862968211706*^9}, {3.6548629982120237`*^9, 3.6548631437354064`*^9}, {3.654863180702627*^9, 3.6548633176224556`*^9}, { 3.6548633631844735`*^9, 3.6548636607821608`*^9}, {3.654863701937386*^9, 3.654863763163146*^9}, {3.654863823937127*^9, 3.654863935089603*^9}, { 3.6548639987373996`*^9, 3.6548640214875393`*^9}, 3.654864097489439*^9, { 3.6548642295882664`*^9, 3.654864235714604*^9}, 3.6548647874605494`*^9}], Cell["\<\ Both conjectures are actually proved here if we accept the paradigmatic \ computational examples here to be parts of the proofs. \ \>", "Text", CellChangeTimes->{{3.654864243087002*^9, 3.654864304407054*^9}, { 3.6548658187398396`*^9, 3.6548658826823845`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["References", "Subtitle", CellGroupingRules->{"TitleGrouping", Inherited}, CellChangeTimes->{ 3.654414521668394*^9, 3.654423026370676*^9, 3.654689334493437*^9, 3.654860065306345*^9, {3.6548624880496006`*^9, 3.654862500299056*^9}, 3.65486253957376*^9}], Cell["\<\ [1] Dainis Zeps. On building 4-critical plane and projective plane \ multiwheels from odd wheels, arXiv:1202.4862v1, [math.CO] Feb 2012 [2] Dainis Zeps. 4-critical wheel graphs of higher order, arXiv:1106.1336v1 [3] Zeps D. Can graph remain k-critical after vertex split?, in preparation, \ 2012 \ \>", "Text", CellChangeTimes->{ 3.654862535907235*^9, {3.6548626985698767`*^9, 3.654862702943537*^9}, { 3.6548640665880747`*^9, 3.6548640671318865`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Here programs that were run:", "Subtitle", CellChangeTimes->{{3.6544230577500563`*^9, 3.654423069672881*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"w1", "=", RowBox[{"WheelGraph", "[", "4", "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"w2", "=", RowBox[{"WheelGraph", "[", RowBox[{"8", ",", RowBox[{"VertexLabels", "\[Rule]", "\"\\""}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"GraphNumShift", "[", RowBox[{"G_", ",", "k_"}], "]"}], ":=", RowBox[{"Graph", "[", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"EdgeList", "[", "G", "]"}], "[", RowBox[{"[", RowBox[{"i", ",", "1"}], "]"}], "]"}], "+", "k"}], ")"}], "\[UndirectedEdge]", RowBox[{"(", RowBox[{ 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