Three-regular Subgraphs of Four-regular Graphs

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1 Europ. J. Combinatorics (1998) 19, Three-regular Subgraphs of Four-regular Graphs OSCAR MORENO AND VICTOR A. ZINOVIEV For an 4-regular graph G (possibl with multiple edges), we prove that, if the number N of distinct Euler orientations of G is such that N 1 (mod 3), then G has a 3-regular subgraph. It gives the new 4-regular graphs with multiple edges which have no 3-regular subgraphs, for which we know the number of Euler orientations. c 1998 Academic Press Limited 1. INTRODUCTION The Berge Sauer conjecture (see [2, 3]) sas that an simple (no multiple edges and loops) 4-regular graph contains a 3-regular subgraph. This conjecture was proved in [4, 6]. In [1, 2] the Chevalle Warning theorem was used to etend this result to graphs with multiple edges, which are 4-regular plus an edge. Our main result, Theorem 2.2, presents the sufficient condition for a 4-regular graph with multiple edges to have a 3-regular subgraph. It gives the new 4-regular graphs with multiple edges which have no 3-regular subgraphs, for which we know eactl the number of Euler orientations. A conjecture b Thomassen [5] sas that an 4-regular connected graph with multiple edges and loops has a 3-regular subgraph, whenever the number of vertices is even. Here we reduce this conjecture to the same conjecture for graphs with onl multiple edges, and one of the authors proved this conjecture [7]. 2. A SUFFICIENT CONDITION We consider the finite undirected graphs G = (V, E) with multiple edges and without loops. A graph G is k-regular if ever verte of G is incident with eactl k edges. DEFINITION 2.1. Let G be a 4-regular graph with the n vertices: V ={v 1,v 2,...,v n } and 2n edges E ={,,...,n }. A Euler orientation of G is a partition E = n i=1 A vi of the set E of edges such that (i) A vi and A v j are disjoint for an i = j and (ii) each A vi has eactl two edges both incident with v i. Let N = N(G) denote the number of distinct Euler orientations of G. THEOREM 2.2. A 4-regular graph G with n vertices has a 3-regular subgraph if N, the number of distinct Euler orientations of G, is such that N 1 (mod 3). PROOF. Let A = (a ij ) be the n m (m = 2n) incidence matri of G : a ij = 1, if verte v i and edge e j are incident and a ij = 0 otherwise. Consider the sstem A X 2 1 X 2 2. X 2 m = = 0, Work supported b NFS grants RII , the component IV of the EPSCoR of Puerto Rico Grant, U.S. Arm Center of Ecellence for Smbolic Methods in Algorithmic Mathematics (ACSAM), of Cornell MSI. Contract DAAL03-91-C-0027 and the Office of Naval Research under grant number N J AMS subject classification code (1991): 05C75, 05C /98/ $25.00/0 ej c 1998 Academic Press Limited

2 370 O. Moreno and V. A. Zinoviev of n quadratic equations in the m variables X 1,...,X m. multi-variable polnomial equations: This is, in fact, a sstem of n F 1 = 0, F 2 = 0,...,F n = 0, (1) where F i, i = 1,...,n, has the form F i = Xi Xi Xi Xi 2 4 and 1 i 1, i 2, i 3, i 4 m. Consider this sstem of equations over the finite field F of order 3. Following the technique developed b Alon et al. [2, 3] suppose now that we have a non-trivial (i.e. not all X i = 0) solution. Let J {1, 2,...,m} be a set of indices j for which X j = 0 and let B be the matri obtained from A b deleting the jth column for each j not in J. Then the graph H whose incidence matri is C (the matri obtained from B b deleting all ero rows) is a 3-regular subgraph of G. The number S of solutions to the sstem above, is such that S = (1 F1 2 )(1 F2 2 ) (1 F2 n ). X 1,...,X m F Now we multipl out the polnomial that appears on the right-hand side and for computations over F (a field of order 3) we have that Xi 3 X i and we can compute the number of solutions modulo 3. In order to obtain onl the trivial solution we must have S 1 (mod 3) and therefore the above sum must also b, when carried out modulo 3. Now (1 F 2 1 ) (1 F2 n ) = A λ G λ (X 1,...,X m ) where A λ F and G λ (X 1,...,X m ) run over all possible monomials in an subset of variables {X1 2,...,X m 2 }. The result now follows easil from the following two points: (1) For the case when the monomial is composed of the full set of variables, A λ is eactl equal to N (mod 3), where N is the number of distinct Euler orientations, as defined before. This follows from the fact that m = 2n is even and for an arbitrar i we have Xi 2 = 1. X i F (2) If G λ is such that it does not contain one of the variables X1 2,...,X m 2 then A λ G λ (X 1,...,X m ) 0 (mod 3). X 1,...,X m F Clearl, Theorem 2.2 can be easil etended as follows. THEOREM 2.3. A 2m-regular graph G has a p-regular subgraph (p is an prime, p 1 m) ifn, the number of distinct Euler orientations of G, is such that N 1 (mod p). EXAMPLE 2.4. The 4-regular graph G has the 3-regular subgraph K 4 (see Fig. 1), but the number of distinct Euler orientations of G is equal to 16. So we conclude that our condition N 1 (mod 3) in Theorem 2.2 is onl sufficient, but not necessar. 3. TRANSFORMATIONS OF 4-REGULAR GRAPHS Now we consider a more general class of graphs. We allow not onl multiple edges as before, but also the loops. Assume that a loop is orientable in two was. Then both Theorems 2.2 and 2.3 are true for graphs with loops without an changes. The following statement follows immediatel.

3 Three-regular subgraphs of four-regular graphs 371 G K 4 FIGURE 1. u u v G 1 G 1 G [0] = G 1 0 G 2 G [1] = G 1 1 G 2 FIGURE 2. LEMMA 3.1. Let G 1 and G 2 be an two graphs with the numbers of distinct Euler orientations N i = N(G i ), i = 1, 2. Define two new graphs G [0] = G 1 0 G 2 and G [1] = G 1 1 G 2 as shown on Fig. 2. Then (a) N(G [0] ) = N 1 N 2 /2 and (b) N(G [1] ) = N 1 N 2. THEOREM 3.2. Let G 1 and G 2 be an 4-regular graphs. Then (a) at least one of the graphs G 1, G 2 and G [0] = G 1 0 G 2 has a 3-regular subgraph; (b) if both graphs G 1 and G 2 have no 3-regular subgraphs, then the graph G [1] = G 1 1 G 2 also has no 3-regular subgraphs. PROOF. (a) If both graphs G 1 and G 2 have no 3-regular subgraphs, then b Theorem 2.2 we have that N(G i ) 1 (mod 3) for i = 1, 2. B Lemma 3.1(a) the graph G [0] has N(G [0] ) = N 1 N 2 /2. But N 1 N 2 /2 1 (mod 3) and, therefore, G [0] has a 3-regular subgraph b Theorem 2.2. Proof (b) follows from the simple observation that in the nontrivial case when G [1] has a 3-regular subgraph the additional verte v belongs to this subgraph which implies that one of G i also has a 3-regular subgraph. Denote b D s (s 1) a 4-regular graph obtained from a ccle with s vertices b replacing ever edge b two parallel edges (D 1 is a point with two loops). It is well known [2] that D s has no 3-regular subgraph, if s is an odd number. B direct calculations we have N(D s ) = 2(2 s 1 + 1). The following statement follows from Lemma 3.1 and Theorem 3.2. THEOREM 3.3. Let l 1,...,l s be the arbitrar natural numbers, s 1. (a) For an 4-regular graph G [0] of tpe D l1 0 0 D ls N(G [0] ) = 2 s (2 l i 1 + 1) (2) i=1 and G [0] has a 3-regular subgraph for all cases, when l 1,...,l s and s are not all odd. (b) If all l i are odd, an graph G [1] of tpe D li 1 1 D ls has no 3-regular subgraph and N(G [1] ) = s (2 l i + 2). (3) i=1

4 372 O. Moreno and V. A. Zinoviev f 1 f 2 G v G v v' v v'' f 3 f 4 e 4 e 3 e 4 e 3 FIGURE 3. LEMMA 3.4. Let G be an 4-regular graph with n 1 vertices and v V (G) be an arbitrar verte. Let G v be obtained from G as shown on Fig. 3. Then both graphs G and G v have or do not have 3-regular subgraphs at the same time. PROOF. It is enough to prove that if an of the graphs has a 3-regular subgraph, then another graph has a 3-regular subgraph also. First assume that G has a 3-regular subgraph, sa H. If none of the edges e i, i = 1, 2, 3, 4, is an edge of H (i.e.,...,e 4 E(H)), then we have nothing to prove: H is a subgraph of G v. So, assume that one of the edges, sa, belongs to E(H). Since H is 3-regular, the other two edges e i and e j, where i, j {2, 3, 4}, i = j, should belong to the set E(H). Then one can easil embed H into the graph H v, a 3-regular subgraph of G v. For eample, if,, e 3 E(H), then the graph H v is obtained from H b adding the vertices v and v, both edges f 1 and f 4 and one of the edges f 2 or f 3. The converse statement follows similarl. For a graph G and arbitrar vertices v 1,...,v u V (G) denote b G v1,...,v u the graph, obtained from G b the same procedure, as shown on Fig. 3. If G has a loop in verte v, which corresponds to the edges and, then G v does not have this loop. It gives the following result. THEOREM 3.5. Let G be an 4-regular graph G with n vertices and u loops, sa v 1, v 2,...,v u, where n u 1. Then a 4-regular graph G v1,v 2,...,v u without loops has or does not have 3-regular subgraphs simultaneousl with the original graph G. REMARK. Theorem 3.5 reduces the conjecture of Thomassen [5] for graphs onl with multiple edges. This conjecture asserts that an 4-regular connected graph with an even number of vertices has a 3-regular subgraph. Now we can sa that this conjecture is correct [7]. REFERENCES 1. N. Alon, S. Friedland, and G. Kalai, Ever 4-regular graph plus an edge contains a 3-regular subgraph, J. Combin. Theor, Ser. B 37 (1984), N. Alon, S. Friedland, and G. Kalai, Regular subgraphs of almost regular graphs, J. Combin. Theor, Ser. B 37 (1984), V. Chvatal, H. Fleischner, J. Sheehan, and C. Thomassen, Three regular subgraphs of four regular graphs, J. Graph Theor 3 (1979), V. A. Tâskinov, Regular subgraphs of regular graphs, Soviet Math. Dokl. 26 (1982), Carsten Thomassen, personal communication, L. Zhang, Ever 4-regular simple graph contains a 3-regular subgraph, Journal of Changsha Railwa Institut (1985),

5 Three-regular subgraphs of four-regular graphs V. A. Zinoviev, On 3-regular subgraphs of 4-regular graphs with multiple edges, (1997), under submission. Received 21 November 1995 and accepted 25 March 1997 O. MORENO Department of Mathematics, Universit of Puerto Rico, Rio Piedras, Puerto Rico, V. A. ZINOVIEV Institute for Problems of Information Transmission, Russian Academ of Sciences, Bol shoi Karetni 19, Moscow, , Russia