A Note on Arboricity of 2-edge-connected Cubic Graphs

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1 Ξ44 Ξ6fi ψ ) 0 Vol.44, No.6 205ff. ADVANCES IN MATHEMATICS(CHINA) No., 205 A Note on Arboricity of 2-edge-connected Cbic Graphs HAO Rongxia,, LAI Hongjian 2, 3, LIU Haoyang 4 doi: 0.845/sxjz b (. Department of Mathematics, Beijing Jiaotong Uniersity, Beijing, 00044, P. R. China; 2. College of Mathematics and System Sciences, Xinjiang Uniersity, Urmqi, Xinjiang, , P. R. China; 3. Department of Mathematics, West Virginia Uniersity, Morgantown, WV , USA; 4. School of Mathematical Sciences, Peking Uniersity, Beijing, 0087, P. R. China) Abstract: The ertex-arboricity a(g) of a graph G is the minimm nmber of sbsets into which the set of ertices of G can be partitioned so that each sbset indces a forest. It is well known that a(g) 3 for any planar graph G, andthata(g) 2 for any planar graph G of diameter at most 2. The conjectre that eery planar graph G withot 3-cycles has a ertex partition (V,V 2) sch that V is an independent set and V 2 indces a forest was gien in [Eropean J. Combin., 2008, 29(4): ]. In this paper, we proe that a 2-edge-connected cbic graph which satisfies some condition has this partition. As a corollary, we get the reslt that eery p-embeddabl-edge-connected cbic graph G (G K 4) has a ertex partition (V,V 2) sch that V is an independent set and V 2 indces a forest. Keywords: edge-connectiity; indced forest; arboricity MR(200) Sbject Classification: 05C0 / CLC nmber: O57.5 Docment code: A Article ID: (205) Introdction Graphs considered in this paper are ndirected, finite and withot loops. The ertexarboricity a(g) (linear ertex-arboricity la(g), respectiely) of a graphg is the minimm nmber of sbsets into which V (G) can be partitioned so that each sbset indces a forest (a linear forest, respectiely). Chartrand, Kronk and Wall [6] first introdced the ertex-arboricity of a graph. They proed that the ertex-arboricity of planar graphs is at most 3. Then the first two athors [5] characterized a planar graph of the ertex-arboricity 3. Yang and Yan [4] showed that a(g) 2 for any planar graphs G of diameter 2. It is proed in [] that eery planar graph G withot triangles at distance less than 2 satisfies a(g) 2, and that eery 2-degenerate graph G has a ertex partition (V,V 2 ) sch that V is an independent set and V 2 indces a forest. Catlin et al. [3] proed that if a connected graph G is neither a cycle nor a cliqe, then there is a coloring of V (G) withatmost (G) 2 colors, sch that each color class indces a forest and one of those indced forests is a maximm indced forest in G. Borodin et al. [2] defined a similar concept for the point arboricity of a graph G. For more reslts abot the ertex-arboricity a(g) of a graph G, the reader is referred to [, 4, 7, 9, 2 3]. Receied date: Reised date: Fondation item: Spported by NSFC (No , No. 7020, No. 2702). rxhao@bjt.ed.cn

2 866 χ ( ffl / 44 The edge-connectiity κ(g) of a graph G is the minimm nmber of edges whose remoal from G reslts in a disconnected graph or triial graph. A graph G is n-edge-connected (n ), if κ(g) n. For an integer k>0, a graph G is called k-reglar, if the degree of eery ertex is k. A 3-reglar graph is also called a cbic graph. Let S be an orientable srface and G be a connected graph. A graph G is said to be embedded in a srface S if it is drawn on S so that edges intersect only at their common end ertices and each component, called a face, ofs \ G is homeomorphic to an open disk. The maximal gens, γ M (G), of a graph G is the maximm gens among the genera of all srfaces S in which G can be embedded. A graph G is called p-embeddable if the maximm gens of G is eqal to β(g) 2,whereβ(G) = E V + is the Beta nmber of G. Let H be a cbic graph, e, E(H) (e and may be the same edge). A new ertex is inserted at each of e and, denoted by and respectiely. Then two new ertices and are added. The graph N = N(, )(H) obtained from H by adding edges, and two doble edges joining and is called an N-extension of H (see Fig. ). e e H G = N(, )(H) e =, G = N(, )(H) Figre N-extension of the graph H Let H be a cbic graph, e,,e 3 E(H) (someofe, and e 3 may be identical). A new ertex is inserted at each of e, and e 3, denoted by, and respectiely, then a new ertex is added. The graph G = M()(H) obtained from H by adding edges, and is called an M-extension of H (see Fig. 2). e e 3 e e 3 H Figr G = M()(H) M-extension of the graph H Let θ i,i =, 2, 3andadmbbell graph be connected graphs depicted respectiely in Fig. 3. A 2-edge-connected cbic graph G is called a doble dmbbell graph if G contains two ertex disjoint indced sbgraphs G[,,,w 4 ]andg[, 2,, ] sch that each of them is a dmbbell graph. For example, Peterson graph is obtained from θ by two M-extensions. The detail is shown in Fig. 4. In [], the following conjectre was gien.

3 6fi Φffi&,»Ωfi, Ψ+: A Note on Arboricity of 2-edge-connected Cbic Graphs 867 Conjectre Eery planar graph G withot 3-cycles has a ertex partition (V,V 2 )sch that V is an independent set and V 2 indces a forest. 2 2 θ θ Figre θ 3 w 4 dmbbell graph θ i,i=, 2, 3 and dmbbell graph = θ H = M(6)(θ ) G = M(0)(H) Peterson graph Figre 4 M-extensions to Peterson graph Althogh there are some reslts on a(g) for planar graphs, few characterizations of the ertex arboricity are known for non-planar graphs. In this paper, we inestigate arboricity for a class of 2-edge-connectedcbic, not necessarily planar graphs. The following reslts are obtained. Theorem 0. Eery 2-edge-connected cbic graph G (G K 4 ) which is not obtained from any doble dmbbell graph by a series of M-extensions or N-extensions has a ertex partition (V,V 2 ) sch that V is an independent set and V 2 indces a forest. The following corollaries follow from Theorem 0.. Corollary 0. If G is a 2-edge-connected cbic graph which is not obtained from any doble dmbbell graph by a series of M-extensions or N-extensions, then a(g) =2. Corollary 0.2 Eery 2-edge-connected cbic p-embeddable graph G (G K 4 )hasa ertex partition (V,V 2 ) sch that V is an independent set and V 2 indces a forest. Corollary 0.3 If G is a 2-edge-connected cbic p-embeddable graph, then a(g) = 2. ProofsofMainReslts For a graph G, ifv (G) can be partitioned into an ordered pair (V,V 2 ) sch that V is an independent set and G[V 2 ] is acyclic (i.e., V 2 indces a forest), then the partition pair (V,V 2 ) is called a good pair of G, whereg[v 2 ] is an indced sbgraph of G by V 2. Note that while K 4 does not hae a good pair, we do hae a(k 4 )=2. Lemma. Let G be a cbic graph that has a good pair (V,V 2), and let G be a cbic graph obtained from G by an M-extension or by an N-extension. Then G also has a good pair. Proof First, sppose that G = M()(G ) is a cbic graph obtained from G by an M- extension (see Fig. 2). Let V = V {} and V 2 = V (G)\V = V 2 {,, }. Since (V,V 2 ) is a good pair of G, by the definition of M-extensions, (V,V 2 ) is a good pair of G.

4 868 χ ( ffl / 44 Second, sppose that G = N(, )(G ) is obtained from G by an N-extension (see Fig. ). Let V = V {} and V 2 = V (G) \ V = V 2 {,,}. Since (V,V 2) is a good pair of G, by the definition of N-extensions, (V,V 2 ) is a good pair of G. Ths this lemma is proed. Lemma.2 Let G be a 2-edge-connected graph. If G is obtained from θ i (i =, 2, 3) by a series of M-extensions or N-extensions, then G has a good pair. Proof By Lemma., it sffices to erify that each of the θ i s has a good pair. Let V = {},V 2 = {}, V 2 = {, },V 22 = { 2, }, V 3 = {, 3, 5 } and V 32 = { 2, 4, 6 }. It is easy to see that V i is an independent set and V i2 indces a forest, ths (V i,v i2 ) is a good pair of θ i for each i {, 2, 3}. Lemma.3 Let G (G K 4 ) be a 2-edge-connected graph. If G is obtained from K 4 by a series of M-extensions or N-extensions, then G has a good pair. Proof Since G K 4, there exists a seqence of cbic graphs G 0,G,,G k, sch that G 0 = K 4 and G k = G, andschthatfori =0,,,k, G i+ is obtained from G i by an M-extension or an N-extension. By Lemma. and the assmption that G K 4, it sffices to erify that G has a good pair. We shall consider the following two cases. Case I G = M()(K 4 ) is a graph obtained from K 4 by an M-extension (see Fig. 2). Let the ertex set of K 4 be {, 2,, } as depicted in Fig. 5. By the definition of an M-extension, three edges e,,e 3 mst be chosen from E(K 4 )toperformthem-extension. By symmetry, G has the following three possibilities. IA 2 2 IC IB 2 2 IC 2 2 IB 2 w2 IC 3 2 Figre 5 One M-extension of K 4 Case IA Sppose that e = = e 3 in this M-extension of K 4. Withot loss of generality, we assme that e = = e 3 = 2, and the reslting graph G after the M-extension is shown in Fig. 5 IA. Case IB Sppose that e = and e 3 e in this M-extension of K 4. Withot loss of generality, we assme that e = = 2. Depending on whether e 3 is adjacent to e or not, we may frther assme that e 3 = (if e and e 3 are adjacent) or e 3 = (if e and e 3 are not adjacent). The reslting graphs G after the M-extension are shown in Fig. 5 IB and IB 2. Case IC Sppose that e,,e 3 in this M-extension are all distinct edges of K 4.ThsH, the sbgraph of K 4 indced by {e,,e 3 }, mst be either a 3-cycle, or a K,3,orapathof3 edges. Withot loss of generality, we assme that e =, = 2 and e 3 = 2 (if H is

5 6fi Φffi&,»Ωfi, Ψ+: A Note on Arboricity of 2-edge-connected Cbic Graphs 869 a3-cycle);ore =, = 2 and e 3 = (if H is a K,3 ); or e = 2, = 2 and e 3 = (if H is a path of 3 edges). The reslting graphs G after the M-extension are shown in Fig. 5 IC,IC 2 and IC 3. In any of Cases IA, IB and IC,letV = {,, } and V 2 = V (G ) V ;inanyofcases IB 2 and IC 2,letV = { 2,, } and V 2 = V (G ) V ;incaseic 3,letV = {,, } and V 2 = V (G ) V. By inspection, in any of these cases, the ordered pair (V,V 2 ) is always a good pair of G. Case II G = N(, )(K 4 ) is a graph obtained from K 4 by one N-extension (see Fig. ) and let the ertex set of K 4 be {, 2,, } as depicted in Fig. 6. By the definition of an N-extension, two edges e, mst be chosen from E(K 4 )toperformthen-extension. By symmetry, G has the following two possibilities (see Fig. 6). 4 2 IIA 2 2 IIB IIB 2 Figre 6 One N-extension of K 4 Case IIA Sppose that e = in this N-extension of K 4. Withot loss of generality, we assme that e = = 2. The reslting graph G after the N-extension is shown in Fig. 6 IIA. Case IIB Sppose that e in this N-extension of K 4. Depending on whether e and are adjacent or not adjacent, we may frther assme that e = 2 and = (if e and are adjacent) or e = 2 and = (if e and are not adjacent). The reslting graphs G after the N-extension are shown in Fig. 6 IIB and IIB 2. In any of these cases, let V = {,, } and V 2 = V (G ) V. By inspection, the ordered pair (V,V 2 ) is always a good pair of G. As in any case, a graph G obtained from K 4 by an M-extension or an N-extension always has a good pair. It follows by Lemma. that G has a good pair. The lemma is proed. The following characterizations for p-embeddabl-edge-connected cbic graphs were gien in [8] and [0] respectiely. Lemma.4 [8] Let G be a 2-edge-connected cbic graph, G is p-embeddable if and only if G satisfies the following conditions: (i) G is obtained from θ by a series of M-extensions or N-extensions, if β(g) is odd. (ii) G is obtained from θ 2 or K 4 by a series of M-extensions or N-extensions, if β(g) is een. Lemma.5 [0] Let G be a 2-edge-connected cbic graph. If G is not p-embeddable, then G is obtained from θ 3 or a doble dmbbell graph by a series of M-extensions or N-extensions. Proof of Theorem 0. Let G (G K 4 ) be a 2-edge-connected cbic graph. If G is p-embeddable, by Lemma.4, G is obtained from θ or θ 2 or K 4 by a series of M-extensions or N-extensions. By Lemmas.2 and.3, G has a good pair (V,V 2 ), i.e., G has a ertex partition (V,V 2 ) sch that V is an independent set and V 2 indces a forest. If G is not p-embeddable, by

6 870 χ ( ffl / 44 Lemma.5 and the assmption of Theorem 0., G is obtained from θ 3 by a series of M-extensions or N-extensions, by Lemma.2, G has a good pair (V,V 2 ), i.e., G has a ertex partition (V,V 2 ) sch that V is an independent set and V 2 indces a forest. Theorem 0. is proed. Corollary 0. is obtained from Theorem 0. and a(k 4 )=2. Corollary 0.2 is obtained from Lemmas.2.4. Corollary 0.3 is a direc corollary of Corollary 0.2 and a(k 4 )=2. References [] Borodin, O.V. and Ianoa, A.O., Planar graphs withot 4-cycles adjacent to 3-cycles are list ertex 2- arborable, J. Graph Theory, 2009, 62(3): [2] Borodin, O.V., Kostochka, A.V. and Toft, B., Variable degeneracy: extensions of Brooks and Gallai s theorems, Discrete Math., 2000, 24(/2/3): 0-2. [3] Catlin, P.A. and Lai, H.-J., Vertex arboricity and maximm degree, Discrete Math., 995, 4(/2/3): [4] Chang, G.J., Chen, C.Y. and Chen, Y.P., Vertex and tree arboricities of graphs, J. Comb. Optim., 2004, 8(3): [5] Chartrand, G. and Kronk, H.V., The point-arboricity of planar graphs, J. Lond. Math. Soc. (), 969, 44(): [6] Chartrand, G., Kronk, H.V. and Wall, C.E., The point-arboricity of a graph, Israel J. Math., 968, 6(2): [7] Chen, Z.Z. and He, X., Parallel complexity of partitioning a planar graph into ertex-indced forests, Discrete Appl. Math., 996, 69(/2): [8] Hang, Y.Q. and Li, Y.P., Extensions on 2-edge connected 3-reglar p-embeddable graphs, Acta Math. Appl. Sin. Engl. Ser., 998, 4(4): [9] Kronk, H.V. and Mitchcm, J., Critical point arboritic graphs, J. Lond. Math. Soc. (2), 975, 9(3): [0] Lü, S.X., On maximm gens and other inariants of graphs, Ph.D. Thesis, Beijing: Beijing Jiaotong Uniersity, 200. [] Raspad, A. and Wang, W.F., On the ertex-arboricity of planar graphs, Eropean J. Combin., 2008, 29(4): [2] Roychodhry, A. and Sr-Kolay, S., Efficient algorithms for ertex arboricity of planar graphs, In: Fondations of Software Technology and Theoretical Compter Science (Thiagarajan, P.S. ed.), Lectre Notes in Compt. Sci., Vol. 026, 995, [3] Škrekoski, R., On the critical point-arboricity graphs, J. Graph Theory, 2002, 39(): [4] Yang, A.F. and Yan, J.J., On the ertex arboricity of planar graphs of diameter two, Discrete Math., 2007, 307(9/20): ? 2-9: 3 B@;>43=5C DJK, HFG 2, 3, IEL 4 (. ffi!λ(χ(%,, 00044; 2. 'flλ(χ(,%" ((-, #μßfl, 'fl, ; 3. $Πffρ*Λ(χ(%, ο±, WV , ν ; 4. Λ(χ( ((-,, 0087) A< G TU Y a(g) w G TSi ff`ξr sgtvu_[ωff`tfi y. }jxφzq^mk G, a(g) 3, ozq^ψafiq Tmk Λ a(g) 2. [Eropean J. Combin., 2008, 29(4): ] Ω]P~fO : q^hλ 3- ptmk WΛ VUT_[ (V,V 2 ) S V wxc`, V 2 ΞR sg. M leq^ 2- Ndztbnr T 3- ±Π G (G K 4 ) WΛ VUT_[ (V,V 2 ) S V wxc`, V 2 ΞR sg. 682 NdzY; ΞRsg; Y