Strengthened Brooks Theorem for digraphs of girth three
SStrengthened Brooks Theorem for digraphs of girththree
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]
December 3, 2018
Abstract
Brooks’ Theorem states that a connected graph G of maximumdegree ∆ has chromatic number at most ∆, unless G is an odd cycleor a complete graph. A result of Johansson [6] shows that if G istriangle-free, then the chromatic number drops to O (∆ / log ∆). In thispaper, we derive a weak analog for the chromatic number of digraphs.We show that every (loopless) digraph D without directed cycles oflength two has chromatic number χ ( D ) ≤ (1 − e − ) ˜∆, where ˜∆ is themaximum geometric mean of the out-degree and in-degree of a vertexin D , when ˜∆ is sufficiently large. As a corollary it is proved that thereexists an absolute constant α < χ ( D ) ≤ α ( ˜∆ + 1) for every˜∆ > Keywords:
Digraph coloring, dichromatic number, Brooks theorem,digon, sparse digraph.
Brooks’ Theorem states that if G is a connected graph with maximum de-gree ∆, then χ ( G ) ≤ ∆ + 1, where equality is attained only for odd cycles ∗ 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. a r X i v : . [ m a t h . C O ] O c t nd complete graphs. The presence of triangles has significant influence onthe chromatic number of a graph. A result of Johansson [6] states that if G is triangle-free, then χ ( G ) = O (∆ / log ∆). In this note, we study thechromatic number of digraphs [3], [8], [11] and show that Brooks’ Theoremfor digraphs can also be improved when we forbid directed cycles of length2. Digraph colorings and the Brooks Theorem
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 acyclic sets. The minimum integer k for which thereexists a k -coloring of D is the chromatic number χ ( D ) of the digraph D. Theabove definition of the chromatic number of a digraph was first introduced byNeumann-Lara [11]. The same notion was independently introduced muchlater by the second author when considering the circular chromatic numberof weighted (directed or undirected) graphs [8]. The chromatic number ofdigraphs was further investigated by Bokal et al. [3]. The notion of chromaticnumber of a digraph shares many properties with the notion of the chromaticnumber of undirected graphs. Note that if G is an undirected graph, and D is the digraph obtained from G by replacing each edge with the pair ofoppositely directed arcs joining the same pair of vertices, then χ ( D ) = χ ( G )since any two adjacent vertices in D induce a directed cycle of length two.Another useful observation is that a k -coloring of a graph G is a k -coloringof a digraph D , where D is a digraph obtained from assigning arbitraryorientations to the edges of G . Mohar [9] provides some further evidence forthe close relationship between the chromatic number of a digraph and theusual chromatic number. For digraphs, a version of Brooks’ theorem wasproved in [9]. Note that a digraph D is k -critical if χ ( D ) = k , and χ ( H ) < k for every proper subdigraph H of D . Theorem 1.1 ([9]) . Suppose that D is a k -critical digraph in which forevery vertex v ∈ V ( D ) , d + ( v ) = d − ( v ) = k − . Then one of the followingcases occurs:1. k = 2 and D is a directed cycle of length n ≥ . k = 3 and D is a bidirected cycle of odd length n ≥ .3. D is bidirected complete graph of order k ≥ . A tight upper bound on the chromatic number of a digraph was firstgiven by Neumann-Lara [11]. 2 heorem 1.2 ([11]) . Let D be a digraph and denote by ∆ o and ∆ i themaximum out-degree and in-degree of D , respectively. Then χ ( D ) ≤ min { ∆ o , ∆ i } + 1 . In this note, we study improvements of this result using the followingsubstitute for the maximum degree. If D is a digraph, we let˜∆ = ˜∆( D ) = max { (cid:112) d + ( v ) d − ( v ) | v ∈ V ( D ) } be the maximum geometric mean of the in-degree and out-degree of thevertices. Observe that ˜∆ ≤ (∆ o + ∆ i ), by the arithmetic-geometric meaninequality (where ∆ o and ∆ i are as in Theorem 1.2). We show that when˜∆ is large (roughly ˜∆ ≥ ), then every digraph D without digons has χ ( D ) ≤ α ˜∆, for some absolute constant α <
1. We do not make an attemptto optimize α , but show that α = 1 − e − suffices. To improve the value of α significantly, a new approach may be required.It may be true that the following analog of Johansson’s result holds fordigon-free digraphs, as conjectured by McDiarmid and Mohar [7]. Conjecture 1.3.
Every digraph D without digons has χ ( D ) = O ( ˜∆log ˜∆ ) . If true, this result would be asymptotically best possible in view of thechromatic number of random tournaments of order n , whose chromatic num-ber is Ω( n log n ) and ˜∆ > (cid:0) − o (1) (cid:1) n , as shown by Erd˝os et al. [4].We also believe that the following conjecture of Reed generalizes to di-graphs without digons. Conjecture 1.4 ([12]) . Let ∆ be the maximum degree of (an undirected)graph G , and let ω be the size of the largest clique. Then χ ( G ) ≤ (cid:24) ∆ + 1 + ω (cid:25) . If we define ω = 1 for digraphs without digons, we can pose the followingconjecture for digraphs. Conjecture 1.5.
Let D be a ∆ -regular digraph without digons. Then χ ( D ) ≤ (cid:24) ∆2 (cid:25) + 1 . Conjecture 1.5 is trivial for ∆ = 1, and follows from Lemma 3.2 for∆ = 2 ,
3. We believe that the conjecture is also true for non-regular digraphswith ∆ replaced by ˜∆. 3 asic definitions and notation
We end this section by introducing some terminology that we will be usingthroughout the paper. The notation is standard and we refer the reader to[2] for an extensive treatment of digraphs. All digraphs in this paper are simple , i.e. there are no loops or multiple arcs in the same direction. We use xy to denote the arc joining vertices x and y , where x is the initial vertex and y is the terminal vertex of the arc xy . We denote by A ( D ) the set ofarcs of the digraph D . For v ∈ V ( D ) and e ∈ A ( D ), we denote by D − v and D − e the subdigraph of D obtained by deleting v and the subdigraphobtained by removing e , respectively. We let d + D ( v ) and d − D ( v ) denote the out-degree (the number of arcs whose initial vertex is v ) and the in-degree (the number of arcs whose terminal vertex is v ) of v in D , respectively. Thesubscript D may be omitted if it is clear from the context. A vertex v issaid to be Eulerian if d + ( v ) = d − ( v ). The digraph D is Eulerian if everyvertex in D is Eulerian. A digraph D is ∆ -regular if d + ( v ) = d − ( v ) = ∆for all v ∈ V ( D ). We say that u is an out-neighbor ( in-neighbor ) of v if vu ( uv ) is an arc. We denote by N + ( v ) and N − ( v ) the set of out-neighbors andin-neighbors of v , respectively. The neighborhood of v , denoted by N ( v ), isdefined as N ( v ) = N + ( v ) ∪ N − ( v ). Every undirected graph G determinesa bidirected digraph D ( G ) that is obtained from G by replacing each edgewith two oppositely directed edges joining the same pair of vertices. If D is adigraph, we let G ( D ) be the underlying undirected graph obtained from D by“forgetting” all orientations. A digraph D is said to be (weakly) connected if G ( D ) is connected. The blocks of a digraph D are the maximal subdigraphs D (cid:48) of D whose underlying undirected graph G ( D (cid:48) ) is 2-connected. A cycle in a digraph D is a cycle in G ( D ) that does not use parallel edges. A directedcycle in D is a subdigraph forming a directed closed walk in D whose verticesare all distinct. A directed cycle consisting of exactly two vertices is calleda digon .The rest of the paper is organized as follows. In Section 2, we improveBrooks’ bound for digraphs that have sufficiently large degrees. In Section3, we consider the problem for arbitrary degrees. ˜∆ The main result in this section is the following theorem.
Theorem 2.1.
There is an absolute constant ∆ such that every digon-freedigraph D with ˜∆ = ˜∆( D ) ≥ ∆ has χ ( D ) ≤ (cid:0) − e − (cid:1) ˜∆ . Lemma 2.2.
Let D be a digraph with maximum out-degree ∆ o , and supposewe have a partial proper coloring of D with at most ∆ o +1 − r colors. Supposethat for every vertex v there are at least r colors that appear on vertices in N + ( v ) at least twice. Then D is ∆ o + 1 − r -colorable.Proof. The proof is easy – since many colors are repeated on the out-neighborhood of v , there are many colors that are not used on N + ( v ). Thus,one can greedily “extend” the partial coloring. Proof of Theorem 2.1.
We may assume that c ˜∆ < d + ( v ) < c ˜∆ and c ˜∆
100 ˜∆ other events A w .Therefore, by the symmetric version of the Local Lemma, it suffices to showthat for each event A v , 4 ·
100 ˜∆ P [ A v ] <
1. We will show that P [ A v ] < ˜∆ − .We do this by proving the following two lemmas. Lemma 2.3. E [ X v ] ≥ e − ˜∆ − .Proof. Let X (cid:48) v be the random variable denoting the number of colors thatare assigned to exactly two out-neighbors of v and are retained by both ofthese vertices. Clearly, X v ≥ X (cid:48) v and therefore it suffices to consider E [ X (cid:48) v ].Note that color i will be counted by X (cid:48) v if two vertices u, w ∈ N + ( v ) arecolored i and no other vertex in S = N ( u ) ∪ N + ( v ) ∪ N ( w ) is assigned color i . This will give us a lower bound on E [ X (cid:48) v ]. There are C choices for color i and at least (cid:0) c ˜∆2 (cid:1) choices for the set { u, w } . The probability that no vertexin S gets color i is at least (1 − C ) | S | ≥ (1 − C ) c ˜∆ . Therefore, by linearityof expectation, we can estimate: E [ X (cid:48) v ] ≥ C (cid:18) c ˜∆2 (cid:19) (cid:18) C (cid:19) (cid:18) − C (cid:19) c ˜∆ ≥ c ( c ˜∆ −
1) exp( − c ˜∆ /C − /C ) ≥ ˜∆ e − Lemma 2.4. P (cid:104) | X v − E [ X v ] | > log ˜∆ (cid:112) E [ X v ] (cid:105) < ˜∆ − .Proof. Let AT v be the random variable counting the number of colors as-signed to at least two out-neighbors of v , and Del v the random variable thatcounts the number of colors assigned to at least two out-neighbors of v butremoved from at least one of them. Clearly, X v = AT v − Del v and thereforeit suffices to show that each of AT v and Del v are sufficiently concentratedaround their means. We will show that for t = log ˜∆ (cid:112) E [ X v ] the followingestimates hold:Claim 1: P [ | AT v − E [ AT v ] | > t ] < e − t / (8 ˜∆) .Claim 2: P [ | Del v − E [ Del v ] | > t ] < e − t / (100 ˜∆) .6he two above inequalities yield that, for ˜∆ sufficiently large, P [ | X v − E [ X v ] | > log ˜∆ (cid:112) E [ X v ]] ≤ e − t
28 ˜∆ + 4 e − t ≤ ˜∆ − log ˜∆ < ˜∆ − , as we require. So, it remains to establish both claims.To prove Claim 1, we use a version of Azuma’s inequality found in [10],called the Simple Concentration Bound. Theorem 2.5 (Simple Concentration Bound) . Let X be a random variabledetermined by n independent trials T , ..., T n , and satisfying the property thatchanging the outcome of any single trial can affect X by at most c . Then P [ | X − E [ X ] | > t ] ≤ e − t c n . Note that AT v depends only on the colors assigned to the out-neighborsof v . Note that each random choice can affect AT v by at most 1. Therefore,we can take c = 1 in the Simple Concentration Bound for X = AT v . Sincethe choice of random color assignments are made independently over thevertices and since d + ( v ) ≤ c ˜∆, we immediately have the first claim.For Claim 2, we use the following variant of Talagrand’s Inequality (see[10]). Theorem 2.6 (Talagrand’s Inequality) . Let X be a nonnegative randomvariable, not equal to 0, which is determined by n independent trials, T , . . . , T n and satisfyies the following conditions for some c, r > :1. Changing the outcome of any single trial can affect X by at most c .2. For any s , if X ≥ s , there are at most rs trials whose exposure certifiesthat X ≥ s .Then for any ≤ λ ≤ E [ X ] , P (cid:104) | X − E [ X ] | > λ + 60 c (cid:112) r E [ X ] (cid:105) ≤ e − λ c r E [ X ] . We apply Talagrand’s inequality to the random variable
Del v . Note thatwe can take c = 1 since any single random color assignment can affect Del v by at most 1. Now, suppose that Del v ≥ s . One can certify that Del v ≥ s by exposing, for each of the s colors i , two random color assignments in7 + ( v ) that certify that at least two vertices got color i , and exposing atmost two other color assignments which show that at least one vertex colored i lost its color. Therefore, Del v ≥ s can be certified by exposing 4 s randomchoices, and hence we may take r = 4 in Talagrand’s inequality. Notethat t = log ˜∆ (cid:112) E [ X v ] >> c (cid:112) r E [ Del v ] since E [ X v ] ≥ ˜∆ /e − E [ Del v ] ≤ c ˜∆. Now, taking λ in Talagrand’s inequality to be λ = t , weobtain that P [ | Del v − E [ Del v ] | > t ] ≤ P [ | Del v − E [ Del v ] | > λ + 60 c (cid:112) r E [ X ]].Therefore, provided that λ ≤ E [ Del v ], we have the confirmed Claim 2.It is sufficient to show that E [ Del v ] = Ω( ˜∆), since λ = O (log ˜∆ (cid:112) ˜∆).The probability that exactly two vertices in N + ( v ) are assigned a particularcolor c is at least c ˜∆ C − (1 − /C ) c ˜∆ ≈ e − , a constant. It remains toshow that the probability that at least one of these vertices loses its coloris also (at least) a constant. We use Janson’s Inequality (see [1]). Let u be one of the two vertices colored c . We only compute the probability that u gets uncolored. We may assume that the other vertex colored c is not aneighbor of u since this will only increase the probability. We show thatwith large probability there exists a monochromatic directed path of lengthat least 2 starting at u . Let Ω = N + ( u ) ∪ N ++ ( u ), where N ++ ( u ) is thesecond out-neighborhood of u . Each vertex in Ω is colored c with probability . Enumerate all the directed paths of length 2 starting at u and let P i be the i th path. Clearly, there are at least ( c ˜∆) such paths P i . Let A i be the set of vertices of P i , and denote by B i the event that all verticesin A i receive the same color. Then, clearly P [ B i ] = (cid:98) ˜∆ / (cid:99) ) ≥ . Then, µ = (cid:80) P [ B i ] ≥ · ( c ˜∆) = 4 c . Now, if δ = (cid:80) i,j : A i ∩ A j (cid:54) = ∅ P [ B i ∩ B j ] inJanson’s Inequality satisfies δ < µ , then applying Janson’s Inequality, withthe sets A i and events B i , we obtain that the probability that none of theevents B i occur is at most e − , and hence the probability that u does notretain its color is at least 1 − e − , as required. Now, assume that δ ≥ µ .The following gives an upper bound on δ : δ = (cid:88) i,j : A i ∩ A j (cid:54) = ∅ P [ B i ∩ B j ] = (cid:88) i,j : A i ∩ A j (cid:54) = ∅ (cid:98) ˜∆ / (cid:99) ) ≤ ( c ˜∆) · c ˜∆ ·
8( ˜∆ − < , for ˜∆ ≥ B i occur is at most e − c / , a constant. Therefore, by linearity of expectation E [ Del v ] = Ω( ˜∆). 8learly, since E [ X v ] ≤ c ˜∆, Lemmas 2.3 and 2.4 imply that P [ A v ] < ˜∆ − .This completes the proof of Theorem 2.1. ˜∆ The bound in Theorem 2.1 is only useful for large ˜∆. Rough estimatessuggest that ˜∆ needs to be at least in the order of 10 . The above approachis unlikely to improve this bound significantly with a more detailed analysis.In this section, we improve Brooks’ Theorem for all values of ˜∆. We achievethis by using a result on list colorings found in [5]. List coloring of digraphs isdefined analogously to list coloring of undirected graphs. A precise definitionis given below.Let C be a finite set of colors. Given a digraph D , let L : v (cid:55)→ L ( v ) ⊆ C be a list-assignment for D , which assigns to each vertex v ∈ V ( D ) a set ofcolors. The set L ( v ) is called the list (or the set of admissible colors ) for v .We say D is L -colorable if there is an L -coloring of D , i.e., each vertex v is assigned a color from L ( v ) such that every color class induces an acyclicsubdigraph in D . D is said to be k -choosable if D is L -colorable for everylist-assignment L with | L ( v ) | ≥ k for each v ∈ V ( D ). We denote by χ l ( D )the smallest integer k for which D is k -choosable.The result characterizes the structure of non L -colorable digraphs whoselist sizes are one less than under Brooks’ condition. Theorem 3.1 ([5]) . Let D be a connected digraph, and L an assignmentof colors to the vertices of D such that | L ( v ) | ≥ d + ( v ) if d + ( v ) = d − ( v ) and | L ( v ) | ≥ min { d + ( v ) , d − ( v ) } + 1 otherwise. Suppose that D is not L -colorable. Then D is Eulerian, | L ( v ) | = d + ( v ) for each v ∈ V ( D ) , and everyblock of D is one of the following: (a) a directed cycle (possibly a digon), (b) an odd bidirected cycle, or (c) a bidirected complete digraph. Now, we can state the next result of this section.
Lemma 3.2.
Let D be a connected digraph without digons, and let ˜∆ =˜∆( D ) . If ˜∆ > , then χ l ( D ) ≤ (cid:100) ˜∆ (cid:101) .Proof. We apply Theorem 3.1 with all lists L ( v ), v ∈ V ( D ) having cardinal-ity (cid:100) ˜∆ (cid:101) . It is clear that the conditions of Theorem 3.1 are satisfied for every9ulerian vertex v . It is easy to verify that the conditions are also satisfiedfor non-Eulerian vertices. Now, if D is not L -colorable, then by Theorem3.1, D is Eulerian and d + ( v ) = (cid:100) ˜∆ (cid:101) for every vertex v . This implies that D is (cid:100) ˜∆ (cid:101) -regular. Now, the conclusion of Theorem 3.1 implies that D con-sists of a single block of type (a), (b) or (c). This means that either D is adirected cycle (and hence ˜∆ = 1), or D contains a digon, a contradiction.This completes the proof.We can now prove the main result of this section, which improves Brooks’bound for all digraphs without digons. Theorem 3.3.
Let D be a connected digraph without digons, and let ˜∆ =˜∆( D ) . If ˜∆ > , then χ ( D ) ≤ α ( ˜∆ + 1) for some absolute constant α < .Proof. We define α = max (cid:110) ∆ ∆ +1 , − e − (cid:111) , where ∆ is the constant inthe statement of Theorem 2.1. Now, if ˜∆ < ∆ then by Lemma 3.2, itfollows that χ ( D ) ≤ (cid:100) ˜∆ (cid:101) ≤ α ( ˜∆ + 1). If ˜∆ ≥ ∆ , then by Theorem 2.1 weobtain that χ ( D ) ≤ (cid:0) − e − (cid:1) ˜∆ ≤ α ( ˜∆ + 1), as required.An interesting question to consider is the tightness of the bound ofLemma 3.2. It is easy to see that the bound is tight for (cid:100) ˜∆ (cid:101) = 2 by consid-ering, for example, a directed cycle with an additional chord or a digraphconsisting of two directed triangles sharing a common vertex. The graphin Figure 1 shows that the bound is also tight for (cid:100) ˜∆ (cid:101) = 3. It is easy toverify that, up to symmetry, the coloring outlined in the figure is the unique2-coloring. Now, adding an additional vertex, whose three out-neighbors arethe vertices of the middle triangle and the three in-neighbors are the remain-ing vertices, we obtain a 3-regular digraph where three colors are requiredto complete the coloring.Another example of a digon-free 3-regular digraph on 7 vertices requiringthree colors is the following. Take the Fano Plane and label its points by1,2,...,7. For every line of the Fano plane containing points a, b, c , take adirected cycle through a, b, c (with either orientation). There is a uniquedirected 3-cycle through any two vertices because every two points line inexactly one line. This shows that the Fano plane digraphs are not isomorphicto the digraph from the previous paragraph. Finally, it is easy to verify thatthe resulting digraph needs three colors for coloring.Note that the digraphs in the above examples are 3-regular tournamentson 7 vertices. It is not hard to check that every tournament on 9 vertices has (cid:100) ˜∆ (cid:101) = 4, and yet is 3-colorable. In general, we pose the following problem.10
12 212
Figure 1: Constructing a 3-regular digraph D with χ ( D ) = 3. Question 3.4.
What is the smallest integer ∆ such that every digraph D without digons with (cid:100) ˜∆( D ) (cid:101) = ∆ satisfies χ ( D ) ≤ ∆ − ? Note that this is a weak version of Conjecture 1.5. By Theorem 2.1,∆ exists. However, we believe that ∆ is small, possibly equal to 4. Thefollowing proposition shows that the above holds for every (cid:100) ˜∆ (cid:101) ≥ ∆ . Proposition 3.5.
Let ∆ be defined as in Question 3.4. Then every digon-free digraph D with (cid:100) ˜∆( D ) (cid:101) ≥ ∆ satisfies χ ( D ) ≤ (cid:100) ˜∆( D ) (cid:101) − .Proof. The proof is by induction on (cid:100) ˜∆ (cid:101) . If (cid:100) ˜∆ (cid:101) = ∆ this holds by thedefinition of ∆ . Otherwise, let U be a maximal acyclic subset of D . Then (cid:100) ˜∆( D − U ) (cid:101) ≤ (cid:100) ˜∆( D ) (cid:101) − U is not maximal. Since we cancolor U by a single color, we can apply the induction hypothesis to completethe proof.As a corollary we get: Corollary 3.6.
There exists a positive constant α < such that for everydigon-free digraph D with (cid:100) ˜∆( D ) (cid:101) ≥ ∆ , χ ( D ) ≤ α (cid:100) ˜∆ (cid:101) .Proof. Let α = max (cid:110) (cid:100) ∆ (cid:101)(cid:100) ∆ (cid:101) +1 , − e − (cid:111) , where ∆ is the constant in thestatement of Theorem 2.1. Now, applying Theorem 2.1 or Proposition 3.5gives the result. 11 eferences [1] N. Alon, J. Spencer, The Probabilistic Method, Wiley, 1992.[2] J. Bang-Jensen, G. Gutin, Digraphs. Theory, Algorithms and Applica-tions, Springer, 2001.[3] D. Bokal, G. Fijavˇz, M. Juvan, P. M. Kayll, B. Mohar, The circularchromatic number of a digraph, J. Graph Theory 46 (2004) 227–240.[4] P. Erd˝os, J. Gimbel, D. Kratsch, Some extremal results in cochromaticand dichromatic theory, J. Graph Theory 15 (1991) 579–585.[5] A. Harutyunyan, B. Mohar, Gallai’s theorem for digraphs, preprint.[6] A. Johansson, Asymptotic choice number for triangle free graphs, DI-MACS Technical Report (1996) 91–95.[7] C. McDiarmid, B. Mohar, private communication, 2002.[8] B. Mohar, Circular colorings of edge-weighted graphs, Journal of GraphTheory 43 (2003) 107–116.[9] B. Mohar, Eigenvalues and colorings of digraphs, Linear Algebra andits Applications 432 (2010) 2273–2277.[10] M. Molloy, B. Reed, Graph Colouring and the Probabilistic Method,Springer, 2002.[11] V. Neumann-Lara, The dichromatic number of a digraph, J. Combin.Theory, Ser. B 33 (1982) 265–270.[12] B. Reed, ω, ∆ , and χχ