Two results on the digraph chromatic number

Transcription

1 Two results o the digraph chromatic umber Ararat Harutyuya Departmet of Mathematics Simo Fraser Uiversity Buraby, B.C. V5A 1S6 aha43@sfu.ca Boja Mohar Departmet of Mathematics Simo Fraser Uiversity Buraby, B.C. V5A 1S6 mohar@sfu.ca March 21, 2011 Abstract It is kow (Bollobás [4]; Kostochka ad Mazurova [13]) that there exist graphs of maximum degree ad of arbitrarily large girth whose chromatic umber is at least c / log. We show a aalogous result for digraphs where the chromatic umber of a digraph D is defied as the miimum iteger k so that V (D) ca be partitioed ito k acyclic sets, ad the girth is the legth of the shortest cycle i the correspodig udirected graph. It is also show, i the same vei as a old result of Erdős [6], that there are digraphs with arbitrarily large chromatic umber where every large subset of vertices is 2-colorable. Keywords: Digraph colorig, dichromatic umber. 1 Digraph Colorigs Let D be a (loopless) digraph. A vertex set A V (D) is called acyclic if the iduced subdigraph D[A] has o directed cycles. A k-colorig of D is a partitio of V (D) ito k acyclic sets. The miimum iteger k for which there exists a k-colorig of D is the chromatic umber χ(d) of the digraph D. This defiitio of the chromatic umber of a digraph was first treated Research supported by FQRNT (Le Fods québécois de la recherche sur la ature et les techologies) doctoral scholarship. Supported i part by a NSERC Discovery Grat (Caada), by the Caada Research Chair program, ad by the Research Grat P of ARRS (Sloveia). O leave from: IMFM & FMF, Departmet of Mathematics, Uiversity of Ljubljaa, Ljubljaa, Sloveia. 1

2 by Neuma-Lara [17]. The same otio was idepedetly itroduced two decades later whe cosiderig the circular chromatic umber of weighted (directed or udirected) graphs [15], see also [3]. This otio of colorigs of digraphs turs out to be the atural way of extedig the theory of udirected graph colorigs sice it provides extesios of most of the basic results from graph colorigs [3, 8, 15, 16]. I this ote we prove, usig stadard probabilistic approach, that two further aalogues of graph colorig results carry over to digraphs. The first result provides evidece that the digraph chromatic umber, like the graph chromatic umber, is a global parameter that caot be deduced from local cosideratios. The secod result, see Theorem 3.1, shows that there are digraphs with large chromatic umber k i which every set of at most c V (D) vertices is 2-colorable, where c > 0 is a costat that oly depeds o k. The aalogous result for digraphs was proved by Erdős [6] with its outcome beig that all sets of at most c are 3-colorable. Both the 3-colorability i Erdős result ad 2-colorability i Theorem 3.1 are best possible. Cocerig the first result, it is well-kow that there exist graphs with large girth ad large chromatic umber. Bollobás [4] ad, idepedetly, Kostochka ad Mazurova [13] proved that there exist graphs of maximum degree at most ad of arbitrarily large girth whose chromatic umber is Ω( / log ). Our Theorem 2.1 provides a extesio to digraphs. The boud of Ω( / log ) from [4, 13] is essetially best possible: a result of Johasso [11] shows that if G is triagle-free, the the chromatic umber is O( / log ). Similarly, Theorem 3.1 is also essetially best possible: it is easy to show that every touramet o vertices has chromatic log umber (1 + o(1)). This follows from the fact that every touramet o vertices cotais a acyclic set of at least log 2 vertices (see for example, [5, 21]). I geeral, it may be true that the followig aalog of Johasso s result holds for digo-free digraphs, as cojectured by McDiarmid ad Mohar [14]. Cojecture 1.1. Every digraph D without digos ad with maximum total degree has χ(d) = O( log ). Theorem 2.1 shows that Cojecture 1.1, if true, is essetially best possible. 2

3 2 Chromatic umber ad girth First, we eed some basic defiitios. Give a loopless digraph D, a cycle i D is a cycle i the uderlyig udirected graph. The girth of D is the legth of a shortest cycle i D, ad the digirth of D is the legth of a shortest directed cycle i D. The total degree of a vertex v is the umber of arcs icidet to v. The maximum total degree of D, deoted by (D), is the maximum of all total degrees vertices i D. It is proved i [3] that there are digraphs of arbitrarily large digirth ad dichromatic umber. Our result is a aalogue of the aforemetioed result of Bollobás [4] ad Kostochka ad Mazurova [13]. Note that the result ivolves the girth ad ot the digirth. Theorem 2.1. Let g ad be positive itegers. There exists a digraph D of girth at least g, with (D), ad χ(d) a / log for some absolute costat a > 0. For sufficietly large we may take a = 1 5e. Proof. Our proof is i the spirit of Bollobás [4]. We may assume that is sufficietly large. Let D = D(, p) be a radom digraph of order defied as follows. For every u, v V (D), we coect uv with probability 2p, idepedetly. Now we radomly (with probability 1/2) assig a orietatio to every edge that is preset. Observe that D has o digos. We will use the value p = 4e, where e is the base of the atural logarithm. Claim 1. D has o more tha g cycles of legth less tha g with probability at least 1 1. Proof. Let N l be the umber of cycles of legth l. The ( ) E[N l ] l!(2p) l l (2p) l ( l 4 )l. Therefore, the expected umber of cycles of legth less tha g is at most g 1. So the probability that D has more tha g cycles of legth less tha g is at most 1/ by Markov s iequality. Claim 2. There is a set A of at most /1000 vertices of D such that (D A) with probability at least 1 2. Proof. As i [4], defie excess degree of D to be ex(d) = d i > (d i ), where d i is the total degree of the i th vertex. Clearly, there is a set of at most ex(d) arcs (or vertices) whose removal reduces the maximum total degree of 3

4 D to. Let X d be the umber of vertices of total degree d, d = 0, 1,..., 1. The ex(d) = 1 d= +1 (d )X d. Now, we estimate the expectatio of X d. By liearity of expectatio, we have: ( 1 E[X d ] d ( e( 1) d ( ) d. 2d ) (2p) d ) d ( ) d 2e Therefore, by liearity of expectatio we have that E[ex(D)] d= +1 d 1 d= +1 1 d= +1 2 ( 1 2 ) = ( ) d 2d ( ) d 1 2d ( ) 1 d 1 2 Now, by Markov s iequality, P[ex(D) > /1000] < 1/2. Let α(d) be the size of a maximum acyclic set of vertices i D. The followig result will be used i the proof of our ext claim ad also i Sectio 3. 4

5 Theorem 2.2 ([22]). Let D D(, p) ad w = p. There is a sufficietly large absolute costat W such that: If p satisfies w W, the, a.a.s. ( ) 2 α(d) (log w + 3e), log q where q = (1 p) 1. Claim 3. Let α(d) be the size of a maximum acyclic set of vertices i D. The α(d) with high probability. 4e log Proof. Sice is sufficietly large, Theorem 2.2 applies ad the result follows. Now, pick a digraph D that satisfies claims 1,2 ad 3. After removig at most /1000+ g /100 vertices, the resultig digraph D has maximum degree at most ad girth at least g. Clearly, α(d ) α(d). Therefore, χ(d ) (1 1/100) 4e log 5e log. 3 Aother result of same ature A result of Erdős [6] states that there exist graphs of large chromatic umber where the iduced subgraph ay costat fractio umber of the vertices is 3-colorable. I particular, it is proved that for every k there exists ɛ > 0 such that for all sufficietly large there exists a graph G of order with χ(g) > k ad yet χ(g[s]) 3 for every S V (G) with S ɛ. The 3-colorability i the aforemetioed theorem caot be improved. A result of Kierstead, Szemeredi ad Trotter [12] (with later improvemets by Nelli [18] ad Jiag [10]) shows that every 4-chromatic graph of order cotais a odd cycle of legth at most 8. We prove the followig aalog for digraphs. Our proof follows the proof of the result of Erdős foud i [1]. Theorem 3.1. For every k, there exists ɛ > 0 such that for every sufficietly large iteger there exists a digraph D of order with χ(d) > k ad yet χ(d[s]) 2 for every S V (D) with S ɛ. Proof. Clearly, we may assume that log k 3 ad k W, where W is the costat i Theorem 2.2. Let us cosider the radom digraph D = D(, p) with p = k2 ad let 0 < ɛ < k 5. 5

6 We first show that χ(d) > k with high probability. Sice k is sufficietly large, Theorem 2.2 implies that α(d) 6 log k/k 2 with high probability. Therefore, almost surely χ(d) 1 6 k2 / log k > k. Now, we show that with high probability every set of at most ɛ vertices ca be colored with at most two colors. Suppose there exists a set S with S ɛ such that χ(d[s]) 3. Let T S be a 3-critical subset, i.e. for every v T, χ(d[t ] v) 2. Let t = T. For every v T, mi{d + D[T ] (v), d D[T ](v)} 2 for otherwise a 2-colorig of D[T ] v could be exteded to D[T ]. Therefore, every vertex i T has total degree of at least 4 i D[T ] which implies that D[T ] has at least 2t arcs. The probability of this is at most ( )( ( 2 t ) ( ) 2) k 2 2t ( e ) ( ) t et(t 1) 2t ( ) k 2 2t t 2t t 2t 3 t ɛ 3 t ɛ ( e 3 tk 4 ) t 4 3 t ɛ ( ) 7tk 4 t ɛ max (1) 3 t ɛ ( ) If 3 t log 2, the 7tk 4 t ( ) 7 log 2 k 4 t ( ) 7 log 2 k 4 3 = o( 1 ( ) ). Similarly, if log 2 t ɛ, the 7tk 4 t (7ɛk 4 ) t ( 7 k )t ( 7 k )log2 = o( 1 ). These estimates ad (1) imply that the probability that χ(d[s]) 2 is o(1). This completes the proof. We show that 2-colorability i the previous theorem caot be decreased to 1 due to the followig theorem. Theorem 3.2. If D is a digraph with χ(d) 3 ad of order, the it cotais a directed cycle of legth o(). Proof. I the proof we shall use the followig digraph aalogue of Erdős- Posa Theorem. Reed et al. [20] proved that for every iteger t, there exists a iteger f(t) so that every digraph either has t vertex-disjoit directed cycles or a set of at most f(t) vertices whose removal makes the digraph acyclic. Defie h() = max{t : tf(t) }. It is clear that h() = ω(1). Let c be the legth of a shortest directed cycle i D. 6

7 If D has h() vertex-disjoit directed cycles, the ch() which implies that c h() = o(). Otherwise, suppose that h() = t. There exists a set S of vertices with S = f(t) such that V (D)\S is acyclic. Sice χ(d) 3, we have that χ(d[s]) 2, which implies that S cotais a directed cycle of legth at most S = f(t) t = h() = o(). Refereces [1] N. Alo, J. Specer, The Probabilistic Method, Wiley, [2] J. Bag-Jese, G. Guti, Digraphs. Theory, Algorithms ad Applicatios, Spriger, [3] D. Bokal, G. Fijavž, M. Juva, P. M. Kayll, B. Mohar, The circular chromatic umber of a digraph, J. Graph Theory 46 (2004) [4] B. Bollobás, Chromatic umber, girth ad maximal degree, Discrete Math. 24 (1978), o. 3, [5] R. C. Etriger, P. Erdős, C. C. Harer, Some extremal properties cocerig trasitivity i graphs, Periodica Mathematica Hugarica 3(1972), [6] P. Erdős, O circuits ad subgraphs of chromatic graphs, Mathematika (1962), 9: [7] P. Erdős, J. Gimbel, D. Kratsch, Some extremal results i cochromatic ad dichromatic theory, J. Graph Theory 15 (1991) [8] A. Harutyuya, B. Mohar, Gallai s Theorem for List Colorig of Digraphs, SIAM Joural o Discrete Mathematics 25(1) (2011), [9] A. Harutyuya, B. Mohar, Stregtheed Brooks Theorem for digraphs of girth three, submitted for publicatio. [10] T. Jiag, Small odd cycles i 4-chromatic graphs, Joural of Graph Theory 37 (2001), [11] A. Johasso, Asymptotic choice umber for triagle free graphs, DI- MACS Techical Report (1996) [12] H. Kierstead, E. Szemeredi, W.T. Trotter, O colorig graphs with locally small chromatic umber, Combiatorica 4 (1984),

8 [13] A.V.Kostochka, N. P. Mazurova, A iequality i the theory of graph colorig, Met Diskret Aaliz 30 (1977), (i Russia). [14] C. McDiarmid, B. Mohar, private commuicatio, [15] B. Mohar, Circular colorigs of edge-weighted graphs, Joural of Graph Theory 43 (2003) [16] B. Mohar, Eigevalues ad colorigs of digraphs, Liear Algebra ad its Applicatios 432 (2010) [17] V. Neuma-Lara, The dichromatic umber of a digraph, J. Combi. Theory, Ser. B 33 (1982) [18] A. Nilli, Short odd cycles i 4-chromatic graphs, Joural of Graph Theory 31 (1999), [19] B. Reed, ω,, ad χ, J. Graph Theory 27 (1998) [20] B. Reed, N. Robertso, P. Seymour, R. Thomas, Packig Directed Circuits, Combiatorica 16(4) (1996) [21] J. Specer, Radom regular touramets, Periodica Mathematica Hugarica 5(1974), [22] J. Specer, C.R. Subramaia, O the size of iduced acyclic subgraphs i radom digraphs, Discrete Mathematics ad Theoretical Computer Sciece, 10:2(2008)