Czechoslovak Mathematical Journal, 48 (123) (1998), Praha
THE MAXIMUM GENUS, MATCHINGS AND THE CYCLE SPACE OF A GRAPH
HUNG-LlN Fu*, Hsinchu, MARTIN SKOVIERA*, Bratislava, and MING-CHUN TSAI*, Hsinchu
(Received October 3, 1995)
Abstract. In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov [5] and a theorem of Nebesky [15].
Keywords: Maximum genus, matching, cycle space MSC 1991: Primary 05C10, secondary 05C70
1. INTRODUCTION
The maximum genus 7M (G) of a connected graph G is the largest integer k such that G has a cellular embedding in the orientable surface of genus k. Formally the maximum genus was introduced by Nordhaus, Stewart and White [16] in 1971, but already in 1970 Khomenko and Yavorsky [11, Section 5] derived a criterion for a graph to have a 2-cell embedding with a single face and by using it calculated the maximum genus of the complete bipartite graphs Kn,n and the n-cubes Qn.
One of the most remarkable facts about the maximum genus is that this topo-logical invariant can be characterized in a purely combinatorial manner. The first combinatorial characterization of the maximum genus is due to Khomenko, Ostro-verkhy and Kuzmenko [10]. Basically the same result was later independently proved by Xuong [21], and an essential part of it by Jungerman [8], It is convenient to state these results in terms of an equivalent quantity called the Betti deficiency of G. It is * Supported in part by the National Science Council of the Republic of China
defined by the equation £(G) = B(G) - 2-yM(G), where /3(G) = |£(G)| - |V(G)| + 1 is the Betti number, i.e., the cycle rank of G.
For a graph H, let c(H), d(H) and b(H) denote the number of components, the number of components with odd number of edges, and the number of components with odd Betti number, respectively.
Theorem 1. (Xuong [21]) Let G be a connected graph. Then £(G) = min {d(G - E(T)); T a spanning tree of G}.
A characterization which is in a sense complementary was subsequently found by Khomenko and Glukhov [9] and independently by Nebesky [13] in a slightly different form. In contrast to Theorem 1, these results express the Betti deficiency as the maximum of a certain combinatorial function.
Let A be a subset of E(G). Set v(G, A) = c(G - A) + b(G -A)- \A\ - 1. Theorem 2. (Nebesky [13]) Let G be a connected graph. Then
Extensions, generalizations and variations on these two theorems were established by various authors, see e.g. [2, 14, 15, 18]. In this paper we pursue the connection between the maximum genus and matchings and prove a characterization of the maximum genus of a graph similar to Theorem 2 by employing the classical theorems due to Hall [7], Tutte [20] and its generalization due to Berge [1].
The relationship between maximum genus and matchings is implicit in the proofs of Theorem 1. By Theorem 1, a connected graph G has zero Betti deficiency (equiva-lently, it admits a single-face 2-cell embedding in some orientable surface) if and only if it has a spanning tree whose cotree consists of components with even number of edges. Such a component can be decomposed into pairs of adjacent edges, and each of the pairs contributes to the maximum genus by one. The adjacent pairs form what is sometimes called an adjacency matching (see, e.g., [6]) and in turn corresponds to a usual matching in a suitably denned graph. In general, the Betti deficiency of a graph equals the minimum number of edges not covered by an adjacency matching. As regards Theorem 2 and similar characterizations, the role of matchings is less obvious. The first step in revealing this connection was taken by Glukhov [5]. To state this result, let B be a basis of the cycle space b(G) of a graph G. Define the graph J(G,B) to have B as its vertex set, with two elements C1 = C2 in B adjacent
if they have a vertex in common. We call J(G, B) the intersection graph of the basis B. Note that J(G,B) is connected whenever G is 2-edge-connected.
Theorem 3. (Glukhov [5, Theorem 1]) A connected graph G has a single-face orientable 2-cell embedding (i.e., £(G) = Q) if and only if for each basis B of the cycle space of G the intersection graph J(G,B) of B has a perfect matching.
As a further step in this direction, Nebesky [15] proved the following. For a tree T of a graph G let G#T be the graph with F(G#T) = E(G) - E(T) and with the property that ef, where e, f E(G) - E(T), forms an edge if T + e + f has only one non-trivial leaf. (Recall that a leaf of G is a 2-edge-connected subgraph maximal with respect to inclusion.) Let u)(H) denote the number of unsaturated vertices of a maximum matching in a graph H.
Theorem 4. (Nebesky [15, Theorem 1]) Let G be a connected graph different from a tree. Then
£(G) = max{w(G#T); T a spanning tree of G}. Moreover, there is a spanning tree YofG such that
Note, however, that every spanning tree T of a graph G determines the standard basis BT of ^(G) where each element of BT uses only one cotree edge. Moreover, one easily derives from the definitions that G#T coincides with J(G, BT). These observations suggest that there should be a common generalization of Glukhov’s Theorem 3 and Nebesky’s Theorem 4. The aim of this paper is to prove such a theorem.
Theorem 5. Let G be a connected graph. Then
C(G) = max{o;(J(G,J3)); B a basis of V(G)}. Moreover, there is a spanning tree YofG such that
It is easy to find a basis for b(G) that is not of the form BT for some spanning tree T of G. It follows that the maximum in Theorem 5 is taken over a larger set than in Theorem 4, and so our main result indeed improves Nebesky’s Theorem 4.
2. DEFINITIONS AND AUXILIARY RESULTS
All graphs considered in this paper are finite and may have loops or multiple edges. A circuit in a graph G is a connected regular subgraph of valency 2, whereas a cycle is a subgraph of G in which every vertex has even valency greater than or equal to 2. The cycle space ^(G) of a graph G is the vector space over the 2-element field spanned by the cycles of G; the sum of two vectors is obtained by taking the symmetric difference of the corresponding sets of edges and omitting all resulting isolated vertices. It follows that the non-zero elements of %?(G) are cycles. The dimension of b(G) is /3(G) = \E(G}\ - \V(G)\ + k, where k denotes the number of components of G. It is called the Betti number of G.
Let G be a connected graph and let T be a spanning tree of G. For a cotree edge e e E(G) - E(T) let T(e) denote the unique cycle in T + e. Then ( T ( e ) ; e e E(G) - E(T)} is a basis for the cycle space of G. As noted above, not every basis of the cycle space can be obtained in this way.
Our results rely on the use of matchings in graphs. We therefore recall some pertinent definitions and theorems.
Let H be a connected loopless graph. A subset M C E(H) will be called a matching if no two edges in M have a vertex in common. A matching with maximum cardinality is called a maximum matching of H and a matching which covers every vertex of H is said to be perfect. The size of a maximum matching in the graph H is its matching number and is denoted by n(H). The number of vertices that are not covered by a maximum matching in H is denoted by u(H). If n is the order of H, then w(H) = n - 2 u , ( H ) .
For a graph H let o(H] denote the number of components of H with odd order. The following generalization of Tutte’s celebrated 1-factor theorem [20] is due to Berge [l].
Theorem 2.1. (Berge) Let H be a, simple graph of order n. Then
Another result which we need is the Konig-Hall theorem [7, 12] about matchings in bipartite graphs. For a vertex x of a graph H let NH(X) denote the neighbourhood of x, the set of vertices adjacent to x. If X C V(H), let NH( X ) = U NH(X).
x6X
Theorem 2.2. (Konig-Hall) Let H be a bipartite graph with bipartition (U, W). Then H contains a matching that covers all vertices in U if and only ifNu(X) ^ \X\ for every subset X C U.
The following lemma is a bridge between the theorems of Berge and Xuong. Recall that the line graph L(H) of a graph H is the graph whose vertices correspond to the edges of H, and where two vertices are joined by an edge if and only if the corresponding edges have an end-vertex in common.
Lemma 2.3. Let G be a connected graph and let T be a spanning tree of G. Then d(G - E(T)) = w(L(G - E ( T ) ) ) .
P r o o f . Consider a component K of the cotree G — E(T) and let K have m edges. For each subset X C E(K) we obviously have d(K — X) < \X\ if m is even, and d(K - X) < \X\ + 1 if m is odd. Since u(L(K)) = max{d(K - X) - \ X \ ; X C E(K)} by Theorem 2.1, we have u(L(K)) = 0 if m is even, and u(L(K)} = 1 if m is odd. (In other words, a connected line graph has either a perfect matching or a matching that misses only one vertex, cf. [3, 19].) Hence,
w(L(G - E ( T ) ) ) = E { w ( L ( K ) ) ; K a component of G - E(T)} = d(G - E(T)}.
We conclude this section with developing a useful technical machinery to handle maximum-genus problems. It is based on the concept of a frame decomposition which was extensively used by Sirafi and Skoviera in [18] within the context of signed graphs. Here, however, we only use the unsigned restriction of this concept.
A pair (F, A) is called a frame decomposition of a connected graph G if F, a frame, is a connected spanning subgraph of G and A = E(G) — E(F). Denote by ol(.F) the number of leaves of F with odd Betti number.
Lemma 2.4. [9, 18] Let (F,A) be a frame decomposition of a connected graph G. Then f (G) > ol(F) - \A\.
A frame decomposition (F, A) is said to be strong if it satisfies the following properties:
(1) every non-trivial leaf R of F is critical, i.e., £(R) = 1 and £(R — e) = 0 for every edge e of R;
(2) (F, A) admits a pairing, i.e., there is an injective mapping which to every edge e e A assigns a non-trivial leaf Re of F such that Re is incident with e.
Lemma 2.5. [18] If (F, A) is a strong frame decomposition of a connected graph G, then £(G) = ol(F) -\A\.
Theorem 2.6. [18] Every connected graph admits a strong frame decomposition. We note in passing that Theorem 2.6 easily implies Theorem 2. Indeed, if (F, A) is a strong frame decomposition of G and A1 = A U / where / is the set of all bridges of F, then i/(G, A') =£(<?)•
3. PROOF OF THEOREM 5
First we show that £(G) > w ( J ( G , B ) ) for every basis B of the cycle space of G. Fix a basis B of b(G) and choose an optimal spanning tree T in G, i.e., one with d(G - E(T)) = £(G). Define a bipartite graph H = H(B,T) with bipartition (U, W) by setting U = B, W = E(G] - E(T) and by joining C 6 U to e 6 W if the edge e belongs to the cycle C. Clearly, \U\ = \W\. We claim that H has a perfect matching. Suppose not. Theorem 2.2 then implies that there exists a subset B1 C B such that
On the other hand, the set D = {T(e); e € E(G) - E(T)} is also a basis for b ( G ) . By elementary linear algebra, D contains a subset D1 with \D'\ ^ |B'| such that every cycle C € B' is a linear combination of elements of D1. Take D1 to have the minimum number of elements. Then any cycle C & B' can be written in the form C = E T ( e ) , where all cycles T(e) appearing in this expression belong to D' and e e E(C) - E(T}. By the minimality of D',
At the same time, the definition of H(B,T) implies that
However, from (l)-(3) we infer that
which is absurd. Thus H(B,T) has a perfect matching. We may therefore denote by Ce the cycle from B matched with the cotree edge e. Obviously, if the cotree edges e and / are adjacent, then Ce and Cf intersect. It follows that the bijection e - Ce provides an isomorphism a of the line graph L(G — E(T)) with a spanning
subgraph of J(G,B). By Lemma 2.3, L(G - E(T)) contains a matching N with £(G) unsaturated vertices. Hence b ( N ) is a matching in J(G,B) and has £(G) unsaturated vertices, too.
Summing up, for every basis B of b ( G ) we have obtained
and so
as well.
To prove the reverse inequality it is sufficient to exhibit a basis B of b ( G ) for which w ( J ( G , B ) ) > £(G). By Theorem 2.6, there is a strong frame decomposition (F, A) of G. We claim that w ( J ( G , BT)) > £(G) for any spanning tree T of the frame F. (Note that T is at the same time a spanning tree of G.)
Fix a spanning tree T of F and set Z = {T(e); e € A}. We estimate w(J(G, BT)). Theorem 2.1 implies that
However, J(G,Br) — Z = J(G - A,Br) is just the disjoint union |J J(R, BTHR) R
which is taken over all leaves R of F, Recall that ol(F) is the number of leaves of F with odd Betti number. It follows that
Since \Z\ = \A\, Lemma 2.5 yields
as claimed. This establishes the first part of our theorem.
In the second part, we again utilize a strong frame decomposition (F, A) of G. So far we have proved that for any spanning tree T of the frame F we have W(J(G,BT)) = £(G). Thus to complete the proof it is sufficient to show that among the spanning trees of F there is one, denoted by Y, that is optimal for G.
By the definition of a strong frame decomposition, (F, A) admits a pairing which to every edge e € A assigns a non-trivial leaf Re such that e is incident with Re.
each e e A choose in Re an edge e' adjacent to e and form the set A' = {e1; e € A}. Now, take Y to be any optimal spanning tree of F — A'.
To show that Y is optimal also for G, we construct a matching in L(G — E ( Y ) ) that has at most ol(F) - \A\ unsaturated vertices. Obviously, the tree YR — Y n R is an optimal spanning tree for any leaf R of F - A'. Every leaf of F — A' is either a leaf 5 of F that does not contain an edge of A' or a leaf of Re - e' for some paired leaf Re of F. In the former case, 5 is critical; hence d(S - E(Ys)) = I and by Lemma 2.3 there is a matching PS of L(S - E(Ys)) with a single unsaturated vertex. In the latter case we have d(Re - e1 - E(Y n Re)) = 0 by the criticality of Re. The same lemma then implies that the line graph of the corresponding cotree has a perfect matching Qe. Taking the union of all the perfect matchings Qe (e € A] with the matchings PS (S an unpaired leaf of F) and with the additional matching {ee'; e G A} we obtain a matching M of L(G — E(Y)}. It is easy to see that M has (at most) one unsaturated vertex per each unpaired leaf S of F and that there are no other unsaturated vertices. Consequently, u(L(G - E ( Y ) ) ) does not exceed the number of unpaired leaves of F, i.e., ol(F) - \A\. Using Lemma 2.5 and Lemma 2.3 again we finally get
implying that d(G - E(Y)) = £(G). Thus Y is an optimal spanning tree of G. The first part of this proof now yields u(J(G,BY)) = e(G) = d(G - E(Y)), and the theorem follows.
4. COROLLARIES
Nebesky's Theorem 3 and Glukhov’s Theorem 4 stated in Introduction are obvious corollaries of our Theorem 5. Here we give some more corollaries of this result. We first restate Theorem 5 in a different form.
Theorem 4.1. Let G be a connected graph. Then the maximum genus of G is
where the minimum is taken over all bases B of the cycle space of G. Moreover, there is a spanning tree T such that JM (G) = u(J(G,
BT))-Here are some corollaries to Theorem 4.1.
Corollary 4.2. [17] Let G he a connected graph. Then 7M(G) = 0 if and only if no two circuits of G have a vertex in common.
Before the next corollary, we need two definitions. A necklace is a graph with vertex set V = {v1,v2, • • • , V2 r] , such that the vertex v2s-i is connected by a single edge to the vertex v2s, s = 1 , . . . , r, and the vertex U2s-2 is connected by a pair of parallel edges to the vertex v2S-1 (where w0 = v2 r), and some loops are added at distinct interior points of those non-multiple edges. Next, a graph G is called a cluster of three cycles if it contains a pair of intersecting circuits C1 and G2 such that V(Ci) n V(C2) induces a path (possibly a single vertex) and G - (E(Ci) U E(G2)) is a path that joins a vertex of C\ to a vertex of C2 and is internally disjoint from C i U C2.
Corollary 4.3. [4] Let G be a 2-edge-connected graph. Then 7M(G) = 1 if and only if it is homeomorphic to a necklace or a cluster of three cycles.
P r o o f . Using Theorem 4.1 it is easy to check that any graph which is home-omorphic to a necklace or to a cluster of three cycles has maximum genus 1. Con-versely, if 7M (G) = I and G is neither homeomorphic to a necklace nor to a clus-ter of three cycles, then there are two pairs of inclus-tersecting circuits, say {Ci,C2} and {C3,G4}, such that Ci, C2, €3 and C4 are linearly independent. Let e,; 6 E(Ci) — U E ( C j ) , where the pairs 61,62 and 63,64 are adjacent. Then for any basis
i&
B of the cycle space of G we can find four members of B, say DI, £>2, D$ and D4 such that e; € E(Di) for each i = 1 , . . . ,4. It follows that {Di,D2} and {D3,D4} are intersecting pairs of cycles in B, whence u ( J ( G , B ) ) > 2. This contradicts The-orem 4.1.
One can go on and ask for a description of graphs having maximum genus 2. However, with Theorem 4.1 and Corollary 4.3 this is easier done than said. Roughly speaking, graphs of maximum genus two are certain combinations of two graphs of the kind described in Corollary 4.3, including a cluster of four or five cycles.
5. CONCLUDING REMARKS
1. The proof of Theorem 5 given above does not depend on Theorem 3 and Theorem 4 which Theorem 5 generalizes. Nonetheless, the inequality £(G) < max w ( J ( G , B)) and the existence of a spanning tree Y with d(G - E(Y)) = £(G) = w(J(G,5y)) can be derived from Theorem 4 since J(G,BY) = G#Y.
2. It seems that the results of this paper might be extended to the maximum Euler genus of a signed graph [18]. A signed graph is a graph whose edges are labelled with signs + and —. An embedding of a signed graph (G,o) in a closed surface, orientable or non-orientable, is an embedding of its underlying graph G where cycles
of G that preserve or reverse orientation of the surface are specified in advance by the signature a. This is done as follows. A cycle (or a circuit) of (G,a) is said to be balanced if the product of signs on its edges is +. In an embedding of ( G , o ) , a circuit of G must be embedded in the surface so as to preserve orientation precisely when it is balanced. This kind of embeddings was studied, e.g., in [18] and [22].
It turns out that signed graph embeddings provide a very natural generalization of embeddings in orientable surfaces where the orientable case is obtained simply by only allowing all-positive signatures. In [18], Siran and Skoviera introduced the maximum Euler genus of a signed graph and proved characterization theorems simi-lar to Theorem 1 (Xuong [21]) and Theorem 2 (Nebesky [13]). Since Theorem 1 and Theorem 2 (or, more precisely, Theorem 2.6 which implies Theorem 2) are crucial to our proofs, there is hope that our present results may be extended to the signed case. However, such an extension is far from immediate. It must include an appro-priate definition of the intersection graph J(G, B) to reflect the balance in the signed graph G and may involve extensions of the Konig-Hall theorem and the Tutte-Berge theorem.
Acknowledgement. This work was done while the second author was visiting
the Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan. He wishes to thank the department for hospitality.
References
[1] C, Berge: Sur le couplage maximum d’un graphe. C. R. Acad. Sci. Paris (A) 247(1958), 258-259.
[2] C. P. Bonnington: The relative maximum genus of a graph. J. Combin. Theory Ser. B 60 (1994), 195-206.
[3] G. Chartrand, A. D. Polirneni and M. J. Stewart: The existence of 1-factors in line graphs, squares, and total graphs. Indag. Math. 35 (1973), 228-232.
[4] J. Chen and J. Gross: Kuratowski-type theorems for average genus. J. Combin. Theory Ser. B 57(1993), 100-121.
[5] A. D. Glukhov: The maximum genus of planar graphs. Ukrain. Mat. Zh. 34 (1982), 97-99. (In Russian.)
[6] J. L. Gross and R. G. Rieper: Local extremain genus-stratified graphs. J. Graph Theory .75(1991), 159-171.
[7] P. Hall: On representation of subsets. J. London Math. Soc. 10 (1935), 26-30. [8] M. Jungerman: A characterization of upper embeddable graphs. Trans. Amer. Math.
Soc. 241 (1978), 401-406.
[9] N. P. Khomenko and A. D. Glukhov: Single-component 2-cell embeddings and the max-imum genus of a graph. Some Topological and Combinatorial Properties of Graphs. Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1980, pp. 5-23. (In Russian.)
[10] N. P. Khomenko, N. A. Ostroverkhy and V. A.Kuzmenko: The maximum genus of a graph. vp-Transformations of Graphs. Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1973, pp. 180-207. (In Ukrainian, English summary.)
[11] N. P. Khomenko and E. V. Yavorsky: p-Transformations of the representation graph. Preprint 70.7, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1970. (In Russian.)
[12] D. Konig: Graphok es matrixok. Math. Fiz. Lapok 38 (1931), 116-119. (In Hungarian.) [13] L. Nebesky: A new characterization of the maximum genus of a graph. Czechoslovak
Math. J. 31 (106) (1981), 604-613.
[14] L. Nebesky: On 2-cell embeddings of graphs with minimum number of regions. Czecho-slovak Math. J. 35 (110) (1985), 625-631.
[15] L. Nebesky: Characterizing the maximum genus of a connected graph. Czechoslovak Math. J. 43 (118) (1993), 177-185.
[16] E. A. Nordhaus, B. M. Stewart and A. T. White: On the maximum genus of a graph. J. Combin. Theory Ser. B 11 (1971), 258-267.
[17] E. A. Nordhaus, R. D. Ringeisen, B. M. Stewart, and A. T. White: A Kuratowski-type theorem for the maximum genus of a graph. J. Combin. Theory Ser. B 12 (1972), 260-267.
[18] J. Sirdn and M. Skoviera: Characterization of the maximum genus of a signed graph. J. Combin. Theory Ser. B 52 (1991), 124-146.
[19] D. P. Sumner: Graphs with 1-factors. Proc. Amer. Math. Soc. 42 (1974), 8-12. [20] W. T. Tutte: The factorization of linear graphs. J. London Math. Soc. 22 (1947),
107-111.
[21] N. H. Xuong: How to determine the maximum genus of a graph. J. Combin. Theory Ser. B (1979), 217-225.
[22] T. Zaslavsky: Orientation embedding of signed graphs. J. Graph Theory 16 (1992), 399-422.
Authors’ addresses: H u n g - L i n Fu and M i n g - C h u n T s a i , Department of Ap-plied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, R.O.C., email: hlfuQmath.nctu.edu.tw; M a r t i n S k o v i e r a , Department of Computer Science, Comenius University, 842 15 Bratislava, Slovakia, email: skoviera@fmph.uniba.sk.
