A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring
of its vertex set such that the colour classes have the same cardinality and
all connected components in the two subgraphs induced by the colour classes
have order at most $k$. Ban and Linial conjectured that {\em every bridgeless
cubic graph admits a $2$--bisection except for the Petersen graph}.
  In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs

Abreu, M.

Goedgebeur, J.

Labbate, D.

Mazzuoccolo, G.

English

arXiv

A -bisection of a bridgeless cubic graph G is a 2-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most k.
Ban and Linial conjectured that every bridgeless cubic graph admits a 2-bisection except for the Petersen graph.

In this note, we prove Ban–Linial’s conjecture for claw-free cubic graphs

ABREU, Marien

LABBATE, Domenico

Mazzuoccolo, G. .

Archivio della Ricerca - Università della Basilicata

A note on 2–bisections of claw–free cubicgraphsMarién Abreu1∗, Jan Goedgebeur2†,Domenico Labbate1∗, Giuseppe Mazzuoccolo3∗1 Dipartimento di Matematica, Università degli Studi della Basilicata,Viale dell’Ateneo Lucano, I-85100 Potenza, Italy.2 Deparment of Applied Mathematics, Computer Science and StatisticsGhent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium.3 Dipartimento di Informatica,Università degli Studi di Verona, Strada le Grazie 15, 37134 Verona, Italy.AbstractA k–bisection of a bridgeless cubic graph G is a 2–colouring of its vertex set suchthat the colour classes have the same cardinality and all connected components inthe two subgraphs induced by the colour classes have order at most k. Ban andLinial conjectured that every bridgeless cubic graph admits a 2–bisection except forthe Petersen graph.In this note, we prove Ban–Linial’s conjecture for claw–free cubic graphs.Keywords: colouring; bisection; claw–free graph; cubic graph.1 IntroductionAll graphs considered in this paper are finite and simple (without loops or multiple edges).Most of our terminology is standard; for further definitions and notation not explicitlystated in the paper, please refer to [4].A bisection of a cubic graph G = (V,E) is a partition of its vertex set V into twodisjoint subsets (B,W) of the same cardinality. We will often identify a bisection of G∗The research that led to the present paper was partially supported by a grant of the group GNSAGAof INdAM and by the Italian Ministry Research Project PRIN 2012 “Geometric Structures, Combinatoricsand their Applications”.†Supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO).Email addresses: marien.abreu@unibas.it (M. Abreu), jan.goedgebeur@ugent.be (J. Goedgebeur),domenico.labbate@unibas.it (D. Labbate), giuseppe.mazzuoccolo@univr.it (G. Mazzuoccolo)1with the vertex colouring of G with two colours B (black) and W (white), such that everyvertex of B and W has colour B and W , respectively. Note that the colouring does notneed to be proper and in what follows we refer to a connected component of the subgraphsinduced by a colour class as a monochromatic component. Following this terminology, wedefine a k–bisection of a graph G as a 2–colouring c of the vertex set V (G) such that:(i) |B| = |W| (i.e. it is a bisection), and(ii) each monochromatic component has at most k vertices.There are several papers in literature considering 2–colourings of regular graphs whichsatisfy condition (ii), but not necessarily condition (i), see [2, 5, 8, 9]. In particular, it iseasy to see that every cubic graph has a 2–colouring where all monochromatic connectedcomponents are of order at most 2. But, in general, such a colouring does not satisfycondition (i), so it is not a 2–bisection. Thus, the existence of a 2–bisection in a cubicgraph is not guaranteed. For instance, the Petersen graph does not admit a 2–bisection.However, the Petersen graph is an exception since it is the only known bridgeless cubicgraph without a 2–bisection. This led Ban and Linial to pose the following conjecture:Conjecture 1.1 (Ban–Linial [3]) Every bridgeless cubic graph admits a 2–bisection,except for the Petersen graph.As far as we know, the best positive result in this direction is given in [7], where it isproven that every cubic graph (not necessarily bridgeless) admits a 3-bisection.A main difficulty in approaching this conjecture is the presence of both local andglobal conditions, that is: conditions (ii) and (i) in the definition of k-bisection, respec-tively. Some standard reductions on the girth and the connectivity of a minimal possiblecounterexample (which can be easily deduced in similar problems) do not work in thiscontext. In particular, so far, we cannot exclude the presence of a 3-cycle or a 2-edge-cutin a minimal counterexample.Recently, we have given a detailed insight into Ban–Linial’s Conjecture (as well as someother related conjectures) in [1], where we also presented theoretical and computationalevidence for it. In particular, Ban–Linial’s Conjecture is proven for bridgeless cubic graphsin the special case of cycle permutation graphs.In this short note, we provide further evidence for it by proving Ban–Linial’s Conjec-ture for claw–free cubic graphs (cf. Theorem 2.2).2 2-bisections of claw-free cubic graphsThe complete bipartite graph K1,3 with colour classes of size 1 and 3, respectively, isusually called the claw graph. A claw–free graph is a graph that does not have a claw asan induced subgraph.2In [6], general claw-free graphs are characterized. In [10], Oum has characterizedsimple, claw-free bridgeless cubic graphs (cf. Theorem 2.1). In order to present such acharacterization we need some preliminary definitions.Let G be a bridgeless claw-free cubic graph. An induced subgraph of G that is iso-morphic to K4 − e is called a diamond and denoted by D in what follows. A string ofdiamonds of G is a maximal sequence D1, ..., Dk of diamonds, in which Di has a vertexadjacent to a vertex of Di+1, 1 ≤ i ≤ k − 1 (see Figure 1). Moreover, if G is a connectedclaw-free simple cubic graph such that each vertex lies in a diamond, then G is called aring of diamonds (see Figure 2).D1 D2 DkFigure 1: A string of diamondsFigure 2: A ring of diamondsTheorem 2.1 (Oum [10]) A graph G is bridgeless claw-free cubic if and only if either(i) G ' K4(ii) G is a ring of diamonds(iii) G can be built from a bridgeless cubic multigraph H by replacing some edges of Hwith strings of diamonds and replacing each vertex of H with a triangle.The main purpose of this section is proving Conjecture 1.1 for the class of claw-freecubic graphs (cf. Theorem 2.2).We recall that a 2–bisection is a bisection (B,W ) such that the connected componentsin the two induced subgraphs G[B] and G[W ] have order at most two. In other words,every induced subgraph is a union of isolated vertices and isolated edges.3Consider a 2-bisection of a claw-free cubic graph and consider the colouring of verticesof a diamond of G. Since any three vertices of a diamond induce a connected subgraph,we have exactly two vertices of the diamond in each colour class. Then, we have onlytwo possibilities: either the two non-adjacent vertices receive different colours (see thediamond D1 in Figure 1) or the same colour (see the diamond D2 in Figure 1). We callthe former colouring asymmetric the latter one symmetric.Theorem 2.2 Let G be a bridgeless claw-free cubic graph. Then, G admits a 2-bisection.Proof It is straightforward that K4 and every ring of diamonds admit a 2-bisection: wecan trivially generalize the example from Figure 2 by colouring each diamond of the ringwith a suitable chosen asymmetric colouring.Consider G as in Theorem 2.1(iii): let H be the underlying bridgeless cubic multigraphand let Habe the graph obtained by replacing each vertex of H with a triangle. ThenG is built from Haby replacing some edges, not in the new triangles, with strings ofdiamonds.First of all, we prove that if Haadmits a 2-bisection, then G admits a 2-bisection. Westart from a 2-bisection of Haand we colour all vertices of the strings of diamonds of Gsuch that we obtain a 2-bisection of G. Let uv be an edge of Hawhich is replaced witha string of diamonds in G. If u and v receive the same colour, say white, in a 2-bisectionof Ha, then we colour the vertices of G corresponding to u and v with the colour white(the same colour they have in Ha), and we colour the vertices of each diamond of thestring with a symmetric 2-colouring such that non-adjacent vertices are black. On theother hand, if u and v receive different colours in the 2-bisection of Ha, then, again, wecolour the vertices of G corresponding to u and v with the same colour they have in Ha,and we colour the vertices of each diamond of the string with an asymmetric 2-colouringby paying attention that adjacent vertices of distinct diamonds receive different colours.Now we prove that Haadmits a 2-bisection. Let M be a perfect matching of H (sinceH is bridgeless, the existence of M is guaranteed by a classical theorem of Petersen, see[11]). The complement of M in H is a 2–factor of H (it could have circuits of length 2, sinceH is a multigraph), say C1, . . . Ct are its circuits. Every edge of M naturally correspondsto an edge of Ha: denote by M ′ the set of those edges in Ha. Every connected componentof the complement of M ′ in Hais isomorphic to a circuit Ci having each vertex replacedby a triangle. Denote by C ′i the circuit corresponding to Ci of length 2|Ci| (cf. Figure 3).We colour the vertices of the even circuit C ′i alternately black and white. Moreover,we colour the ends of every edge of M ′ with different colours. It is straightforward thatsuch a 2-colouring is a bisection. Furthermore, every vertex of a circuit C ′i is adjacent toat least two vertices of the opposite colour and every vertex not in a circuit C ′i is adjacentto exactly two vertices of the opposite colour (its neighbour in M ′ and one of its furtherneighbours).4C′iFigure 3: Pictures of C ′i in Ha. Edges of M ′ in bold.References[1] M. Abreu, J. Goedgebeur, D. Labbate, G. Mazzuoccolo. Colourings of cubic graphsinducing isomorphic monochromatic subgraphs. ArXiv:1705.06928v1, preprint 2017,(submitted).[2] N. Alon, G. Ding, B. Oporowski, D. Vertigan. Partitioning into graphs with onlysmall components. J. Combin. Theory Ser. B, 87 (2003), 231–243.[3] A. Ban, N. Linial. Internal Partitions of Regular Graphs. J. Graph Theory, 83 (2016),no. 1, 5–18.[4] J.A. Bondy and U.S.R. Murty. Graph Theory. Springer Series: Graduate Texts inMathematics, 244 (2008).[5] R. Berke, T. Szabó. Relaxed two–coloring of cubic graphs. J. Combin. Theory Ser.B, 97 (2007), no. 4, 652–668.[6] M. Chudnovsky, P.D. Seymour. The structure of claw-free graphs. Surveys in com-binatorics, 327 (2005), 153–171.[7] L. Esperet, G. Mazzuoccolo, M. Tarsi. Flows and bisections in cubic graphs. Toappear in J. Graph Theory, DOI:10.1002/jgt.22118.[8] P. Haxell, T. Szabó, G. Tardos. Bounded size components-partitions and trasversals.J. Combin. Theory Ser. B, 88 (2003), 281–297.[9] N. Linial, J. Matousek, O. Sheffet, G. Tardos. Graph colouring with no largemonochromatic components. Combin. Probab. Comput. , 17 (2008), no. 4, 577–589.[10] S. Oum. Perfect Matchings in Claw-free Cubic Graphs. Electron. J. Combin., 18(2011), P.62.[11] J. Petersen. Die Theorie der regulären graphs, Acta Mathematica, 15 (1891), 193–220.5

A note on 2--bisections of claw--free cubic graphs.

A k-bisection of a bridgeless cubic graph G is a 2-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most k. Ban and Linial conjectured that every bridgeless cubic graph admits a 2-bisection except for the Petersen graph. In this note, we prove Ban–Linial's conjecture for claw-free cubic graphs

A note on 2--bisections of claw--free cubic graphs

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