Table Of ContentSpringer Proceedings in Mathematics & Statistics
Frederick Hoffman Editor
Combinatorics,
Graph Theory
and Computing
SEICCGTC 2020, Boca Raton, USA,
March 9–13
Springer Proceedings in Mathematics &
Statistics
Volume 388
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Frederick Hoffman
Editor
Combinatorics, Graph
Theory and Computing
SEICCGTC 2020, Boca Raton, USA,
March 9–13
Editor
FrederickHoffman
DepartmentofMathematicalSciences
FloridaAtlanticUniversity
BocaRaton,FL,USA
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedingsinMathematics&Statistics
ISBN 978-3-031-05374-0 ISBN 978-3-031-05375-7 (eBook)
https://doi.org/10.1007/978-3-031-05375-7
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Preface I
The Southeastern International Conference on Combinatorics, Graph Theory, and
Computing (SEICCGTC) is an international meeting of mathematical scientists,
heldannuallyinMarch,duringSpringBreakatFloridaAtlanticUniversity(FAU)
inBocaRaton,Florida.Theconferenceincludesaprogramwithplenarylecturesby
invitedspeakers,aswellassessionsofcontributedpaperseachday.Inaddition,two
or three invitational special sessions are offered each year. A valuable part of the
conferenceistheopportunityaffordedforinformalconversationsaboutthemethods
participantsemployintheirprofessionalworkinbusiness,industry,andgovernment
andabouttheircurrentresearch.
The51stmeetingwasheldinthenewlyrenovatedStudentUnionbuildingatFAU,
March9–13,2020.Fivedistinguishedresearchers,atvariousstagesoftheircareers,
accepted invitations to attend as plenary speakers at this year’s 51st SEICCGTC:
PierreBaldi,UniversityofCalifornia,Irvine,USA;PavolHell,SimonFraserUniver-
sity,Canada;PatriciaHersh,UniversityofOregon,USA;PanosM.Pardalos,Univer-
sity of Florida, USA; and Kai-Uwe Schmidt, Paderborn University, Germany. Dr.
Pardaloshadtocancelhistalkatthelastmoment,duetoillnessandthepandemic.
Eachoftheotherplenaryspeakersgavetwotalks.Thereweretwospecialsessions
thisyear,oneonResearchbyWomeninGraphTheoryanditsApplications,orga-
nized by Leslie Hogben, and one on Extremal Graph Theory, organized by Neal
Bushaw.Bothwerewellattendedandwellreceived.Plenaryandcontributedtalks
coveredawidevarietyoftopicsincluding:newtoolsforcountingandlinearprogram-
ming,usingtopologicalmethods;graphhomomorphism;graphswithloops;highly
non-linearfunctionsandcodingtheory;associationschemes;deeplearningandits
mathematicalfoundations;extremalgraphtheory;posets;latinsquares;combinato-
rialgames;coloring,connectivity,domination,labeling,andpartitioningofgraphs;
alongwithassociatedalgorithmsandapplications.
Thecoronaviruspandemicof2020createdsomedifficultiesforourconference.
Oneplenaryspeakercouldnotattend,andweexperienced20cancellationsduetothe
virusoutbreak.Severalparticipantshadtheirreturntravelplansdisrupted;afewwere
quiteworriedforadayortwo.TheFAUDepartmentofMathematicalScienceshosted
approximately150conferenceparticipantsandguests,markinganothersuccessful
v
vi PrefaceI
meetingoftheSEICCGTC!Forthemostpart,the51stmeetingoftheSEICCGTC
wasagreatsuccessandconferenceparticipantsexpressedtheirapprovaloftheoverall
qualityofspeakersandprogramsandthecontinuousimprovementsinthetechnology
provided.Wehavecausetocelebrate!
OnTuesday,March10,2020,wecelebratedthepublicationofthecommemorative
bookentitled,“50YearsofCombinatorics,GraphtheoryandComputing,”published
byCRC/Taylor&Francis.Twenty-ninepastplenaryspeakersandpastconference
participants contributed 21 chapters for the book, edited by Fan Chung, Ronald
Graham, Ronald Mullin, Frederick Hoffman, Douglas West and Leslie Hogben.
The Institute of Combinatorics and its Applications held its annual meeting on
Wednesday,March11.
TheconferencealsofeaturedanoutdoorreceptionMondayeveningontheLive
Oak Pavilion Patio, a sumptuous beachfront banquet Wednesday evening at the
DelraySandsResort,aswellanexcursionThursdayafternoontotheFlaglerMuseum
in Palm Beach, followed by an informal reception Thursday evening at the brand
newSchmidtFamilyComplex.Thesocialprogramwascappedoffbyawonderful
Survivors’PartyFridayevening,hostedbyAaronMeyerowitzandAndreaSchuver
attheirhome!
This year, we took another step toward elevating the quality of the conference,
by agreeing to publish our conference proceedings with Springer Nature, in their
PROMSseriesofhardbackconferenceproceedings.Thepurposeistomoreeffec-
tivelyandefficientlycontinuetodisseminateimportantadvancesintherepresented
disciplines and to ensure that the conference continues to promote better under-
standingoftherolesofmodernappliedmathematics,combinatoricsandcomputer
science;demonstratethecontributionofeachdisciplinetotheothers;anddecrease
gaps between the fields, as it did through fifty years of publishing in the journal,
CongressusNumerantium.
The conference was supported by the Department chair and staff, with tech-
nical support by Andrew Gultz. Outside support came from the National Security
Agency,SpringerNature,CRCPress/Taylor&Francis,Algorithms,andTheInstitute
of Combinatorics and its Applications. Conference coordination and organization
wassuperblyprovidedbyDr.MariaProvost.
I gratefully acknowledge the support and assistance of Sara Heuss Holliday,
RichardLow,ZviRosen,FarhadShahrokhi,andJohnWiermaninthecompilation
andreviewingoftheseProceedings.Wealsothankallourreferees.
BocaRaton,USA FrederickHoffman
Preface II
Ratio balancing numbers, introduced here by Jeremy Bartz and his coauthors, are
a generalization of balancing numbers, a concept from number theory involving
triangularnumbers.Theauthorsdefinetheconceptandpresentexamples,existence
results,andconjectures.
BohanQuandStephenJ.Curranshowthatthenumberβ=(bb−1−1)/(b−1)2,
whereb≥3,hasseveralinterestingmultiplicativeproperties.Inthebasebnumber
system,β =(123···(b−4)(b−3)(b−1)) .Theyshowthatthedigitsofthenumber
b
Kβ,forintegersK suchthat1≤K ≤(b−1)2,asanumberinthebasebnumber
systemcan begenerated fromanarithmeticsequence reduced modulo b −1with
anappropriateadjustment.
DennisDavenportandhiscoauthorsreportonrecentresultsoftheirresearchgroup
onRiordanarrays.TheygeneralizeaknownrowconstructionofRiordanarraysto
aresultonthedeterminationofdoubleRiordanarrays.
TimothyMyersconstructstheCliffordgraphalgebraforanywindmillgraphW(r,
m),whichconsistsofmcopiesofthecompletegraphK adjoinedatonecommon
r
vertex; and for any Dutch windmill graph Dm which consists of m copies of the
r
r-cyclegraphC adjoinedatonecommonvertex.Hethenappliestheconstruction
r
to give a new proof that these graphs, which possess the friendship property, are
preciselythefriendshipgraphs.
PaulPeartandFrancoisRamarosonconstructandfindthevaluesforcertainchar-
actersumsinvolvingquadraticcharacters.Themethodisnewandemployselliptic
curves.Detailedproofsareprovided.
In work that originated in an REU at Illinois State University, Joel Jeffries and
hiscoauthorsinvestigateamultigraphGwiththeunderlyingstructureofa4-cycle
whereeachedgemultiplicityintheset{1,2,3,4}isrepresented.Theyrefertoeach
ofthethreesuchmultigraphsasaStanton4-cycle.ForeachsuchG,theyconsiderλ
suchthatthereexistsaG-decompositionofλK .
n
BrigitteServatiusconsidersthek-planematroid,whichisamatroidontheedge
set,I,ofabipartitegraph,H=(A,B;I),definedbyacountingcondition.Sheshows
that2k-connectivityofH impliesthatI isaspanningsetforthek-planematroidon
vii
viii PrefaceII
the edge set of the complete bipartite graph on (A, B). For k = 2 she explains the
connectionstorigidityintheplaneandtoconjecturesofWhiteley.
FarhadShahrokhiderivesanupperboundonthetracefunctionofahypergraphH
andgivessomeapplications.Forinstance,anewupperboundfortheVCdimension
of H, or vc(H), follows as a consequence and can be used to compute vc(H) in
polynomialtimeprovidedthatH hasboundeddegeneracy.Thiswasnotpreviously
known, and improves computing time in some cases. Another consequence is a
generallowerboundonthedistinguishingtransversalnumberofHthatgivesriseto
applicationsindominationtheoryofgraphs.
SarahHeussHollidaycontinuesworkonaquestionraisedin2017byHedetniemi:
ForwhichgraphsGdoestheindexedfamilyofopenneighborhoodshaveasystem
of distinct representatives? In earlier work with collaborators, that question was
answered,andnecessaryconditionsandassociatedparameterswereexplored.Haenel
andJohnsonlookedoverlongestpathsandcycles.Theworkherefurthergeneralizes
anddeepenstheirexaminations.
AtifAbueidaandKennethRobleeexamineharmoniouslabelingsofstarliketrees.
It has been shown using cyclic groups that the disjoint union of an odd cycle on s
verticesandstarliketreeswiththecentralvertexadjacenttosomeeventmanys-paths
is harmonious. They consider the disjoint union of an odd cycle with at least two
starlike trees with new notions of harmonious labelings to accommodate the case
where|V|>|E|.
AmeancoloringofaconnectedgraphGoforder3ormoreisanedgecoloringof
Gwithpositiveintegerssuchthatthemeanofthecolorsoftheedgesincidentwith
everyvertexisaninteger.Theassociatedcolorofavertexisitschromaticmean.If
distinctverticeshavedistinctchromaticmeans,thentheedgecoloringisarainbow
mean coloring of G. In their paper, Ebrahim Salehi and his coauthors investigate
rainbowmeancoloringsoftrees.
Peg solitaire is a game in which pegs are placed in every hole but one, and the
playerjumpsoverpegstoremovethem.In2011,thisgamewasgeneralizedtographs.
Here,RobertA.BeelerandAaronD.Grayexaminegraphsinwhichanysingleedge
addition changes solvability. They provide necessary and sufficient conditions for
solvabilityforacertainfamily.Theyshowthatinfinitesubsetsofthisfamilyareedge
criticalanddeterminethemaximumnumberofpegsthatcanbeleftonthisfamily
withtheconditionthatajumpismadewheneverpossible.Finally,theygivealistof
graphsoneightverticesthatareedgecritical.
Asetofvertices,S,inastronglyconnecteddigraphD,issplitdominatingprovided
it is: (1) dominating and (2) D − S is trivial or not strongly connected. The split
domination number is the minimum cardinality of a split dominating set for that
digraph.SarahMerzandhercoauthorsshowthatforanyk-regulartournament,the
split domination number is at least (2k+3)/3 and this bound is tight. They explore
propertiesofregulartournamentswithsplitdominationnumberequaltothelower
bound,includingsufficientconditionsfor{1}-extendability.
DavidR.Prierandhiscoauthorsgiveindependenceanddominationresultsforsix
chess-likepiecesontriangularboardswithtriangularspacesandtriangularboards
PrefaceII ix
withhexagonalspaces.Thequestionofindependenceanddominationforthesesame
boardsonthesurfaceofatetrahedronisintroduced,andsomeinitialresultsaregiven.
AgraphhasanefficientdominatingsetifthereexistsasubsetofverticesDsuch
thateveryvertexinthegraphisdominatedbyexactlyonevertexinD.LylePaskowitz
andhiscoauthorsinvestigateefficientdominationonthestackedversionsofeachof
the eleven Archimedean Lattices, and determine the existence or non-existence of
efficientdominatingsetsoneachlatticethroughintegerprogramming.Theproofsof
existenceareconstructive,andtheproofsofnon-existencearegeneratedbyinteger
programs. They find efficient dominating sets on seven of the stacked lattices and
provethatnosuchsetsexistontheotherfourstackedlattices.
Let G be a graph with vertex setV (G) and edge setE(G). A (p; q)-graph G=
(V;E)issaidtobeAL(k)-traversalifthereexistasequenceofverticesv ,v ,...,v
1 2 p
suchthatforeachi=1,2,...,p–1,thedistanceforv andv isequaltok.Wecalla
i i+1
graphGak-stepsHamiltoniangraphifithasaAL(k)-traversalinGandthedistance
betweenv andv isk.AgraphGissaidtobehereditaryk-stepshyperhamiltonian
p 1
ifitisk-stepsHamiltonianandforanyvinG,thevertex-deletedsubgraphG\{v}
isalsok-stepsHamiltonian.Inthispaper,Hsin-haoSuandhiscoauthorsinvestigate
subdivisiongraphsofawheelgraphandC ×K toseewhichare2-stepsHamiltonian
4 2
andhereditarynon2-stepsHamiltonian.
LetGbeagraphwithaveragedegreegreaterthank–2.Erdo˝sandSósconjectured
that G contains every tree on k vertices as a subgraph. The circumference of the
graphG,c(G),isthenumberofedgesonalongestcycle.GilbertandTinerproved
thatifc(G)isatmostk,thenGcontainseverytreeonkvertices.HereA.M.Heissan
andGaryTinerimprovethisresultandshowthatthe Erdo˝s-Sósconjectureholdsfor
graphswhosecircumferenceisatmostk+1.
YoshimiEgawaandKenjiKimuraconsiderarelationshipbetweenaregulargraph
and a regular factor of its vertex-deleted subgraph. Katerinis proved that if r is an
even integer and k is an integer with 1 ≤ k ≤ r/2, and G is an r-regular, r-edge-
connectedgraphofoddorder,thenG\{x}hasak-factorforeachx∈V (G).When
theresult“foreachx ∈V (G)”ofKaterinisisreplaced“forsomex ∈V (G),”they
considerwhatconditioncanhold.Onemainresultis:Letr andk beevenintegers
suchthat4≤k≤r/2,and(cid:4)beaminimumintegersuchthat(cid:4)≥r/(r−2k+4),and
Gbeanr-regular,2(cid:4)-edge-connectedgraphofoddorder.Then,thereissomex∈V
(G) such that G \{x} has a k-factor. Moreover, if r ≥ 4k − 8, then we can replace
2(cid:4)-edge-connectedwith2-edge-connected.
In his paper, LeRoy B. Beasley gives several definitions of connectedness and
extendibilityofpathsandcyclesindirectedgraphs.Hedefinessetsofdigraphsby
varioustypesofconnectednessorextendibilityandgivessomecontainmentsaswell
asexamplestoshowpropercontainment.
Extraconnectivity generalizes the concept of connectivity of a graph but it is
moredifficulttocompute.Inhispaper,EddieChengandhiscoauthorscomputethe
g-extraconnectivityofthearrangementgraphforsmallg(g ≤6)withthehelpofa
computerprogram.Inaddition,theyprovideanasymptoticresultforgeneralg.
