Two results on the digraph chromatic number
aa r X i v : . [ m a t h . C O ] O c t Two results on the digraph chromatic number
Ararat Harutyunyan ∗ Department of MathematicsSimon Fraser UniversityBurnaby, B.C. V5A 1S6email: [email protected]
Bojan Mohar †‡ Department of MathematicsSimon Fraser UniversityBurnaby, B.C. V5A 1S6email: [email protected]
August 7, 2018
Abstract
It is known (Bollob´as [4]; Kostochka and Mazurova [12]) that thereexist graphs of maximum degree ∆ and of arbitrarily large girth whosechromatic number is at least c ∆ / log ∆. We show an analogous resultfor digraphs where the chromatic number of a digraph D is definedas the minimum integer k so that V ( D ) can be partitioned into k acyclic sets, and the girth is the length of the shortest cycle in thecorresponding undirected graph. It is also shown, in the same vein asan old result of Erd˝os [5], that there are digraphs with arbitrarily largechromatic number where every large subset of vertices is 2-colorable. Keywords:
Chromatic number, digraph, digraph coloring, dichromaticnumber, girth.
Let D be a (loopless) digraph. A vertex set A ⊂ V ( D ) is called acyclic ifthe induced subdigraph D [ A ] has no directed cycles. A k -coloring of D is apartition of V ( D ) into k or fewer acyclic sets. The minimum integer k forwhich there exists a k -coloring of D is the chromatic number χ ( D ) of the ∗ Research supported by FQRNT (Le Fonds qu´eb´ecois de la recherche sur la nature etles technologies) doctoral scholarship. † Supported in part by an NSERC Discovery Grant (Canada), by the Canada ResearchChair program, and by the Research Grant P1–0297 of ARRS (Slovenia). ‡ On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana,Ljubljana, Slovenia. k in which everyset of at most c | V ( D ) | vertices is 2-colorable, where c > k . The analogous result for digraphs was proved by Erd˝os[5] with its outcome being that all sets of at most cn are 3-colorable. Boththe 3-colorability in Erd˝os’ result and 2-colorability in Theorem 3.1 are bestpossible.Concerning the first result, it is well-known that there exist graphs withlarge girth and large chromatic number. Bollob´as [4] and, independently,Kostochka and Mazurova [12] proved that there exist graphs of maximumdegree at most ∆ and of arbitrarily large girth whose chromatic number isΩ(∆ / log ∆). Our Theorem 2.1 provides an extension to digraphs.The bound of Ω(∆ / log ∆) from [4, 12] is essentially best possible: aresult of Johansson [10] shows that if G is triangle-free, then the chromaticnumber is O (∆ / log ∆). Similarly, Theorem 3.1 is also essentially best pos-sible: Erd˝os et al. [6] showed that every tournament on n vertices haschromatic number O ( n log n ).In general, it may be true that the following analog of Johansson’s resultholds for digon-free digraphs, as conjectured by McDiarmid and Mohar [13]. Conjecture 1.1.
Every digraph D without digons and with maximum totaldegree ∆ has χ ( D ) = O ( ∆log ∆ ) . Theorem 2.1 shows that Conjecture 1.1, if true, is essentially best possi-ble.
First, we need some basic definitions. For an extensive treatment of di-graphs, we refer the reader to [2]. Given a loopless digraph D , a cycle in D
2s a cycle in the underlying undirected graph. The girth of D is the length ofa shortest cycle in D , and the digirth of D is the length of a shortest directedcycle in D . The total degree of a vertex v is the number of arcs incidentto v . The maximum total degree of D , denoted by ∆( D ), is the maximumof all total degrees of vertices in D . The out-degree and the in-degree of avertex v are denoted by d + ( v ) and d − ( v ), respectively.It is proved in [3] that there are digraphs of arbitrarily large digirthand dichromatic number. Our result is an analogue of the aforementionedresult of Bollob´as [4] and Kostochka and Mazurova [12]. Note that the resultinvolves the girth and not the digirth. Theorem 2.1.
Let g and ∆ be positive integers. There exists a digraph D of girth at least g , with ∆( D ) ≤ ∆ , and χ ( D ) ≥ a ∆ / log ∆ for some absoluteconstant a > . For ∆ sufficiently large we may take a = .Proof. Our proof is in the spirit of Bollob´as [4]. We may assume that ∆ issufficiently large.Let D = D ( n, p ) be a random digraph of order n defined as follows. Forevery u, v ∈ V ( D ), we connect uv with probability 2 p , independently. Nowwe randomly (with probability 1/2) assign an orientation to every edge thatis present. Observe that D has no digons. We will use the value p = ∆4 en ,where e is the base of the natural logarithm. Claim 1. D has no more than ∆ g cycles of length less than g with probabilityat least − .Proof. Let N l be the number of cycles of length l in D . Then E [ N l ] ≤ (cid:18) nl (cid:19) l !(2 p ) l ≤ n l (2 p ) l ≤ ( ∆4 ) l . Therefore, the expected number of cycles of length less than g is at most∆ g − . So the probability that D has more than ∆ g cycles of length less than g is at most 1 / ∆ by Markov’s inequality. Claim 2.
There is a set A of at most n/ vertices of D such that ∆( D − A ) ≤ ∆ with probability at least .Proof. Let X d be the number of vertices of total degree d , d = 0 , , ..., n − excess degree of D to be ex ( D ) = P n − d =∆+1 ( d − ∆) X d . Clearly, there is a set of at most ex ( D ) arcs (or vertices) whoseremoval reduces the maximum total degree of D to at most ∆.3ow, we estimate the expectation of X d . By linearity of expectation, wehave: E [ X d ] ≤ n (cid:18) n − d (cid:19) (2 p ) d ≤ n (cid:18) e ( n − d (cid:19) d (cid:18) ∆2 en (cid:19) d ≤ n (cid:18) ∆2 d (cid:19) d . Therefore, by linearity of expectation we have that E [ ex ( D )] ≤ n − X d =∆+1 nd (cid:18) ∆2 d (cid:19) d ≤ n ∆2 n − X d =∆+1 (cid:18) ∆2 d (cid:19) d − ≤ n ∆2 n − X d =∆+1 (cid:18) (cid:19) d − ≤ n ∆2 · ( ) ∆ − = n · ∆2 ∆ ≤ n . Now, by Markov’s inequality, P [ ex ( D ) > n/ < / . Let α ( D ) be the size of a maximum acyclic set of vertices in D . Thefollowing result will be used in the proof of our next claim and also inSection 3. Theorem 2.2 ([20]) . Let D ∈ D ( n, p ) . There is an absolute constant W such that if p satisfies np ≥ W , then, a.a.s. α ( D ) ≤ (cid:18) q (cid:19) (log np + 3 e ) , here q = (1 − p ) − . Claim 3.
Let D ∈ D ( n, p ) . Then α ( D ) ≤ en log ∆∆ with high probability.Proof. Since ∆ is sufficiently large, Theorem 2.2 applies and the result fol-lows.Now, pick a digraph D that satisfies claims 1, 2 and 3. After removing atmost n/ g ≤ n/
100 vertices, the resulting digraph D ∗ has maximumdegree at most ∆ and girth at least g . Clearly, α ( D ∗ ) ≤ α ( D ). Therefore, χ ( D ∗ ) ≥ n (1 − / en log ∆ / ∆ ≥ ∆5 e log ∆ . A result of Erd˝os [5] states that there exist graphs of large chromatic numberwhere every induced subgraph with up to a constant fraction number of thevertices is 3-colorable. In particular, it is proved that for every k there exists ǫ > n sufficiently large there exists a graph G of order n with χ ( G ) > k and yet χ ( G [ S ]) ≤ S ⊂ V ( G ) with | S | ≤ ǫn .The 3-colorability in the aforementioned theorem cannot be improved.A result of Kierstead, Szemeredi and Trotter [11] (with later improvementsby Nilli [17] and Jiang [9]) shows that every 4-chromatic graph of order n contains an odd cycle of length at most 8 √ n .We prove the following analog for digraphs. Our proof follows the ideasof Erd˝os found in [1]. Theorem 3.1.
For every k , there exists ǫ > such that for every sufficientlylarge integer n there exists a digraph D of order n with χ ( D ) > k and yet χ ( D [ S ]) ≤ for every S ⊂ V ( D ) with | S | ≤ ǫn .Proof. Clearly, we may assume that log k ≥ k ≥ √ W , where W is theconstant in Theorem 2.2. Let us consider the random digraph D = D ( n, p )with p = k n and let 0 < ǫ < k − .We first show that χ ( D ) > k with high probability. Since k is sufficientlylarge, Theorem 2.2 implies that α ( D ) ≤ n log k/k with high probability.Therefore, almost surely χ ( D ) ≥ k / log k > k. Now, we show that with high probability every set of at most ǫn verticescan be colored with at most two colors. Suppose there exists a set S with | S | ≤ ǫn 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 | . Since D [ T ] is 3-critical,every v ∈ T satisfies min { d + D [ T ] ( v ) , d − D [ T ] ( v ) } ≥ D [ T ] − v could be extended to D [ T ]. This implies that D [ T ] has at least2 t arcs. The probability of this event is at most X ≤ t ≤ ǫn (cid:18) nt (cid:19)(cid:18) (cid:0) t (cid:1) t (cid:19) (cid:18) k n (cid:19) t ≤ X ≤ t ≤ ǫn (cid:16) ent (cid:17) t (cid:18) et ( t − t (cid:19) t (cid:18) k n (cid:19) t ≤ X ≤ t ≤ ǫn (cid:18) e tk n (cid:19) t ≤ ǫn max ≤ t ≤ ǫn (cid:18) tk n (cid:19) t (1)If 3 ≤ t ≤ (log n ) , then (7 tk /n ) t ≤ (7(log n ) k /n ) t ≤ (cid:0) n ) k /n (cid:1) = o (1 /n ) . Similarly, if (log n ) ≤ t ≤ ǫn , then (cid:0) tk /n (cid:1) t ≤ (7 ǫk ) t ≤ (7 /k ) t ≤ (7 /k ) (log n ) = o (1 /n ) . These estimates and (1) imply that the probability that χ ( D [ S ]) ≤ o (1). This completes the proof.The 2-colorability in the previous theorem cannot be decreased to 1 dueto the following theorem. Theorem 3.2. If D is a digraph with χ ( D ) ≥ and of order n , then itcontains a directed cycle of length o ( n ) .Proof. In the proof we shall use the following digraph analogue of Erd˝os-Posa Theorem. Reed et al. [19] proved that for every integer t , there existsan integer f ( t ) so that every digraph either has t vertex-disjoint directedcycles or a set of at most f ( t ) vertices whose removal makes the digraphacyclic. Define h ( n ) = max { t : tf ( t ) ≤ n } . It is clear that h ( n ) = ω (1).Let c be the length of a shortest directed cycle in D and let t := h ( n ).If D has t vertex-disjoint directed cycles, then ct ≤ n which implies that c ≤ nh ( n ) = o ( n ). Otherwise, there exists a set S of vertices with | S | ≤ f ( t )such that V ( D ) \ S is acyclic. Since χ ( D ) ≥
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