Brualdi, Richard A.

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Title2-MULTIGRAPHS(Algebraic Combinatorial Theory)Author(s)Brualdi, Richard A.Citation数理解析研究所講究録 (1988), 671: 116-125Issue Date1988-09URL http://hdl.handle.net/2433/100819RightType Departmental Bulletin PaperTextversionpublisherKyoto University$1A\tilde{\neg}$2-MULTIGRAPHSRichard A. BrualdiDepartment of MathematicsUniversity of WisconsinMadison, WI 53706 USAThis lecture is based on the two papers [1] and [2] written jointly with T. S. Michael. Werefer to these articles for the references not provided here.Let $G=$ (V,E) be a $u\sim t$ raph. Thus V $=\{1,2,\ldots,n\}$ is a set of $n$ vertices and E4is a multiset of unordered pairs of distinct vertices called $ed-\underline{\alpha es}$ . In a $2- multi_{\underline{ff}1}\cdot aph$ each edgeoccurs at most twice in $E$ , equivalently at most two edges join each pair of distinct vertices.(1) $\underline{Exam_{-}\mathfrak{v}1e}$Let $d_{i}$ equal the number of edges that meet vertex $i$ . Then $d_{1}=4$ , $d_{2}=4$ , $d_{3}=4$ ,$d_{4}=3,$ $d_{5}=3$ , and $d_{6}=2$ . The degree $\underline{seo_{-}uence}$ is (4,4,4,3,3,2).With proper labelling of the vertices we may always assume that the degree sequence$D=(d_{1}, \ldots,d_{n})$ of a multigraph satisfies $d_{1}\geq\cdots\geq d_{n}$ . This assumption is made implicitlythroughout.Our object of study is the class $\mathfrak{G}_{2}(D)$ of all 2-multigraphs with the same degreesequence $D=(d_{1}, \ldots,d_{n})$ . The frst question that arises is that of the nonemptiness of thisclass.I671 1988 116-125$\triangleleft\cdot A\backslash \gamma_{-\neg}’$.(2) $\underline{Theorem}$ (Chungphaisan 1974). There exists -a 2-multigraph $\underline{with}$ degree se uence$D=(d_{1}, \ldots,d_{n})$ if $\underline{and}\underline{onlv}\mathscr{Q}$(3) $d_{1}+-+d_{n}\llcorner s\underline{even}$ ,(4) $\sum^{k}d_{i}\leq 2k(k-1)+\sum^{n}$ $\min\{2k,d_{i}\}$ $(k=1, \ldots,n)$ .$i=1$ $i=k+1$The above is a special case of a more general theorem of Chungphaisan for r-multigraphsobtained by replacing the three $2’ s$ above by $r’ s$ . The case $r=1$ is then the following wellknow result.(5) Theorem (Erd\"os/Gallai 1960). $\underline{There}$ $\underline{exists}$ a $g\llcorner apk$ $\underline{with}$ degree $\ovalbox{\tt\small REJECT} se$$D=(d_{1}, \ldots,d_{n})\underline{if}\underline{and}\underline{onlv}\underline{if}$(6) $d_{1}+\cdots+d_{n}\underline{is}\underline{even}$ ,(7) $\sum_{i=1}^{k}d_{i}\leq k(k-1)+\sum_{i=k+1}^{n}\min\{k,d_{i}\}$ $(k=1, \ldots,n)$ .We now adopt the following color convention for 2-multigraphs. An edge of multiplicity2 is denoted by one blue $gdA_{rightarrow}^{e}$ (designated by a bold edge). An edge of multiplicity 1 isdenoted by one $xe\Delta\underline{e}\ovalbox{\tt\small REJECT}\underline{e}$ :$=_{j}1$ : $\frac{blue}{1}j$$\ovalbox{\tt\small REJECT}_{j}1$ : $\frac{red}{1}$ .2$A^{=}\backslash \#rightarrow\prime P$Thus a 2-multigraph is a graph in which every edge is colored red or blue, where redcounts as 1 and blue counts as 2.(8) $\underline{Exam_{-}o}$le. Two 2-multigraphs with the degree sequence $D=$ (4,4,4,4,4,):The example on the right is parsimonious in the sense that it has the smallest number ofcolored edges among all 2-multigraphs in $\mathfrak{G}_{2}(D)$ .Let $G$ be a 2-multigraph in $\mathfrak{G}_{2}(D)$ . Then$\beta(G)=$ number of blue edges of $G$ ,$\rho(G)=$ number of red edges of $G$ ,$\tau(G)=$ number of colored edges of $G$ ,$m(G)=$ number of edges of $G$ counting multiplicites.It follows that(9) $\tau(G)=\beta(G)+\rho(G)$ ,(10) $m(G)=(d_{1}+\cdots+d_{n})/2$$=2\beta(G)+\rho(G)$$=\tau(G)+\beta(G)$$=2\tau(G)-p(G)$ .3$\gamma A$/, $\backslash$$C^{\backslash }$}The 2-multigraph $G$ is called -oarsimonions provided $\tau(G)$ $\leq$ $\tau(H)$ for all 2-multigraphs $H$ with the same degree sequence $D$ as G. It follows from (10) that each ofthe following is equivalent to the parsimony of $G$ :$\beta(G)\geq\beta(H)$ for all $H\in \mathfrak{G}_{2}(D)$ ,$\rho(G)\leq p(H)$ for all $H\in \mathfrak{G}_{2}(D)$ .(11) $\underline{Exam_{-}n1e}$ . The following two 2-multigraphs with degree sequence (3,3,2,1,1) are$parsmonous:\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$The red $L^{ra}4^{\underline{h}}$ of a 2-multigraph is the graph (not the 2-multigraph) determined by itsred edges with isolated vertices deleted.(12) Theorem. $\underline{Each}\underline{connected}\underline{com_{-}Donent}$ of $\underline{the}\underline{red}$ grap$\underline{h}\underline{of}\underline{a}\underline{oarsimonious}- 2- multi\underline{\not\subset}ra_{-}Dh$i\S $\underline{either}g\underline{star}$ $(\underline{a} K_{1,t} \underline{for}\underline{some} t\geq 1)$ $gr\underline{a}$ $\underline{triana}$-le $(\underline{a} K_{3})$ . $\underline{Furthermore}\underline{at}\underline{most}\underline{o}ng$connected $\underline{com_{-}D}$onent is a $\underline{trian}-\alpha le$ .(13) $\ovalbox{\tt\small REJECT} m$. The converse of Theorem (12) does not hold. The 2-multigraph$\lambda_{t})\dagger_{i}$$\lambda$$_{4}8:$:with degree sequence $D=$ (2,2,1,1,1,1) has a red graph consisting of two stars $K_{1,2}$ , butthe 2-multigraph with degree sequence $D$I $[$has fewer colored edges.A $\Omega^{a}a\Delta X$ is a graph each of whose connected components is a star or a triangle with atmost one component equal to a triangle. A stellar $galaAx$ is a galaxy with no triangle. ByTheorem (12) the $\underline{red}r\infty ak\underline{of}$ a -oarsimonious $2-\underline{multi}-\alpha ra_{-}oh$ is a salaxv.(14) $\underline{Remark}$ . The Erd\"os/Gallai Theorem gives necessary and sufficient conditions that theparsimonious 2-multigraphs in a class $\mathfrak{G}_{2}(D)$ have empty red graphs (i.e. all blue edges).These conditions are(15) $d$ is even $(i=1, \ldots,n)$ ,$i$(16) $d_{1}+\cdots+d_{n}\equiv 0(mod 4)$ ,(17) $\sum^{k}d_{i}\leq 2k(k-1)+\sum^{n}$ $\min\{2k,d_{i}\}$ $(k=1, \ldots,n)$ .$i=1$ $i=k+1$$\zeta$$L-\mathfrak{n}\wedge$$\cdot$(The $d_{i}$ clearly have to be even. Now apply the Erd\"os/Gallai Theorem to the degreesequence $(d_{1}/2, \ldots,d_{n}/2)$ and color all edges of the re ulting graph blue.) Condition (17) isthe Chungphaisan condition (4). If we retain the condition (15), the only other possibility for$d_{1}+\cdots+d_{n}$ is(18) $d_{1}+\cdots+d_{n}\equiv 2$ (mod 4).Since the $d_{i}$ are even, the red graph of a parsimonious 2-multigraph cannot contain anystars. It follows that (15), (17), and (18) are necessary and sufficient conditions in order thatthe parsimonious 2-multigraphs in $\mathfrak{G}_{2}(D)$ have red graphs consisting only of a red triangle.For the two cases discussed in Remark (14) there are algorithms to construct aparsimonious 2-multigraph in a nonempty class $\mathfrak{G}_{2}(D)$ . In the case that (15), (16), and (17).are satisfied it suffices to apply the well-known algorithm of Havel and Hakimi to the degreesequence $(d_{1}/2, \ldots, d_{n}/2)$ and then to color all edges of the resulting graph blue. In the casethat (15), (17), and (18) are satisfied the algorithm of Havel and Hakimi can be refined toproduce a parsimonious 2-multigraph (the problem is to decide where the red edges of thetriangle go).(19) $\underline{Problem}$. Find an algorithm to construct a parsimonious 2-multigraph in a nonemptyclass $\mathfrak{G}_{2}(D)$ .We now consider the question: What kinds of red graphs [galaxies] can occur in theparsimonious 2-multigraphs in $\underline{one}$ class $\mathfrak{G}_{2}(D)$?A collection $g$ of galaxies is $class- com_{-}Datible$ provided there is a class $\mathfrak{G}_{2}(D)$ such thatfor each $H\in Q$ there is a parsimonious 2-multigraph $G_{H}\in \mathfrak{G}_{2}(D)$ whose red graph isisomorphic to H.122(20) ma. The galaxies of a class-compatible collection 9 of galaxies contain(21) The same number $\rho$ of edges (by parsimony),(22) The same number $\mu$ of vertices of odd degree (the number of odd components of thedegree sequence D).If we denote by $\Delta$ ( $=0$ or 1) the number of triangles in a galaxy, then the number of oddstars [stars whose center has odd degree] equals $\mu-\rho+3\Delta$ . But we do not know whether thegalaxies $i_{J}\iota$ a class-compatible collection all have the same value of $\Delta$ (thus we do not knowwhether they have the same number of odd stars). We do however have the following result.(23) $\ovalbox{\tt\small REJECT} ore$ Let $g\underline{be}\underline{a}\underline{collection}$ of stellar galaxies $havin\underline{\not\subset}$ the same number $\rho$ Of $\underline{e}4*es$$\underline{and}\underline{the}$ same number $\mu$ $\underline{of}$ vertices $\underline{of}\underline{odd}$ degree (eouivalentlv the same number $\underline{of}$ oddstars). $\underline{Then}9\underline{is}\underline{class- com_{-}natible}$.(24) $\ovalbox{\tt\small REJECT} 1e$. The three galaxies below (with $\rho=7$ and $\mu=8$ ) are class-compatible:(25) Qu stio . Are the following two galaxies (one of which has a triangle and the other ofwhich does not) class compatible?’7$tt$ ’. :..The preceding theory has an analogue for bipartite 2-multigraphs with prescribed degreesequence. The vertices of a bipartite 2-multigraph are partitioned into two sets X and $Y$with each edge joining vertices from different sets. We let $R=$ $(r_{1}, \ldots, r_{m})$ and$S=(s_{1}, \ldots,s_{n})$ be the degree sequences of the vertices in the two parts. Let $\_{2}(R,S)$ denotethe class of bipartite 2-multigraphs with these degree sequences $R$ and S. The class$\_{2}(R,S)$ corresponds to the class $\mathfrak{U}_{2}(R,S)$ of all (0,1,2)-matrices with row sum vector $R$and column sum vector $S$ (use the reduced adjacency matrix).(26) $\underline{Exam_{-}o1e}$ .$y_{1}\backslash$ $y_{2}$ $y_{3}$$x_{2}^{1}x_{3}x$$\{\begin{array}{lll}2 0 20 l 0l 2 2\end{array}\}$$R=(4,1,5)$ , $S=(3,3,4)$The notion of parsimony carries over to the class $\mathfrak{B}_{2}(R,S)$ . In terms of $\mathfrak{U}_{2}(R,S)$ weobtain that a matrix A in $\mathfrak{U}_{2}(R,S)$ is parsimonious provided it has the largest number of$O’ s$ (equivalently the smallest number of l’s, equivalently the largest number of $2’ s$ ) amongall matrices in $\mathfrak{U}_{2}(R,S)$ .Arguments analogous to those for $\mathfrak{G}_{2}(D)$ allow one to conclude that the red graphs ofparsimonious bipartite 2-multigraphs have their connected components equal to stars.Because of the bipartite assumption there can be no triangle, however a different difficultyarises. Some stars $K_{1,t}(t>1)$ have their center in X (corresponding in the matrix$L^{g,}A^{Y}$.formulation to l’s in the same row) while other stars $K_{t,1}$ $(t>1)$ have their center in $Y$(corresponding to l’s in the same column). The former are called $\underline{row}\underline{stars}$ while the latterare called $cA!4mn\underline{stars}$. Then there are the ambi uous (or unoriented) stars $K$1,1(corresponding to a 1 which is the only 1 in its row $\underline{and}$ the only 1 in its column.(27) $\underline{E_{Xam_{-}D}1e}$.$K_{1,2}$ (rowstar)$\{\begin{array}{llll}1 1 1 1 l 1\end{array}\}$$rightarrow K_{1,1}$ (ambiguous star)We use the concepts of a galaxy and of class-compatibility of galaxies in an analogousway to that used above. If a collection of galaxies is class-compatible then they have(28) the same number $\rho$ of l’s (edges),(29) the same number $\lambda$ of rows with an odd number of l’s,(30) the same number $\nu$ of columns with an odd number of 1’s.In addition since $\lambda+\nu-\rho$ equals the number of odd stars (includes ambiguous stars), theyhave(31) the same number $\lambda$ of odd stars.9$\sim^{-\wedge\prime;\not\subset_{\}}}i\wedge:\vee r,$,(32) $\underline{Theorem}$ . A collection of $\underline{\alpha}-laxies\underline{with}\underline{the}\underline{same}$ $\rho,$ $\lambda$ , $\underline{and}$ $\nu$ is class-com atible$\infty rovide$ it $\underline{is}-\underline{oossible}\underline{to}$ assign $g\underline{n}$ orientation (row $\underline{or}\underline{column}$) $\underline{to}$ each ambi uous star of $\underline{ea}sh$$arrow alaxr$ su $\underline{that}$ each galaxv $\underline{has}\underline{the}$ same number gf $\underline{odd}$ row $\underline{stars}\underline{and}\underline{the}$ samg number $\underline{of}\underline{odd}$column stars.The preceding theorem may be true without the orientation assumption.(33) $\frac{E_{XamD}1e}{-}$ The following three galaxies have $\rho=5$ , $\lambda=3$ and $\nu=3$ but do not $s$atisfythe orientation assumption in Theorem (3.2).1 odd 1 odd 1 odd starcolumn star row star (row or column?)Are these galaxies class-compatible?REFERENCES1. Richard A. Brualdi and T. S. Michael, The class of 2-multigraphs with a prescribeddegree sequence, Linear and Multilinear Algebra, to appear.2. Richard A. Brualdi and T. S. Michael, The class of matrices of $O’ s$ , l’s, and $2’ s$ withprescribed row and column sums, Linear Algebra and $i\underline{ts}A_{-}D_{-}D$]$ications$ , to appear.1 $\sigma$

2-MULTIGRAPHS(Algebraic Combinatorial Theory)

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