Covering 2-colored complete digraphs by monochromatic d-dominating digraphs
aa r X i v : . [ m a t h . C O ] F e b Covering 2-colored complete digraphs by monochromatic d -dominating digraphs Louis DeBiasio ∗ , Andr´as Gy´arf´as † February 26, 2021
Abstract
A digraph is d -dominating if every set of at most d vertices has a common out-neighbor. For all integers d ≥
2, let f ( d ) be the smallest integer such that the verticesof every 2-edge-colored (finite or infinite) complete digraph (including loops) can becovered by the vertices of at most f ( d ) monochromatic d -dominating subgraphs. Notethat the existence of f ( d ) is not obvious – indeed, the question which motivated thispaper was simply to determine whether f ( d ) is bounded, even for d = 2. We answerthis question affirmatively for all d ≥
2, proving 4 ≤ f (2) ≤ d ≤ f ( d ) ≤ d (cid:16) d d − d − (cid:17) for all d ≥
3. We also give an example to show that there is no analogousbound for more than two colors.Our result provides a positive answer to a question regarding an infinite analogue ofthe Burr-Erd˝os conjecture on the Ramsey numbers of d -degenerate graphs. Moreover,a special case of our result is related to properties of d -paradoxical tournaments. Throughout this note a directed graph (or digraph for short) is a pair (
V, E ) where V can befinite or infinite and E ⊆ V × V (so in particular, loops are allowed). A digraph is complete if E = V × V . For v ∈ V , we write N + ( v ) = { u : ( v, u ) ∈ E } and N − ( v ) = { u : ( u, v ) ∈ E } . For a positive integer k , we define [ k ] := { , . . . , k } . Note that regardless of whether G = ( V, E ) is a graph or a digraph, if H = ( V ′ , E ′ ) and V ′ ⊆ V and E ′ ⊆ E , we will write H ⊆ G and we will always refer to H as a subgraph of G rather than making a distinctionbetween “subgraph” and “subdigraph.”Let G = ( V, E ) be a digraph. For
X, Y ⊆ V we say that X dominates Y if ( x, y ) ∈ E for all x ∈ X, y ∈ Y . We say that G is d -dominating if for all S ⊆ V with 1 ≤ | S | ≤ d , S dominates some w ∈ V . Note that it is possible for w ∈ S , in which case we must have( w, w ) ∈ E . Reversing all edges of a d -dominating digraph gives a d -dominated digraph.These notions are well studied for tournaments (see Section 3).A cover of a digraph G = ( V, E ) is a set of subgraphs { H , . . . , H t } such that V ( G ) = S i ∈ [ t ] V ( H i ). By a 2 -coloring of G = ( V, E ), we will always mean a 2-coloring of the edges of G ; i.e. a function c : E → [2]. Given a 2-coloring of G , we let E i be the set of edges receivingcolor i (i.e. E i = c − ( { i } )) and G i = ( V, E i ) for i ∈ [2]. A cover of G by monochromaticsubgraphs is a cover { H , . . . , H t } of G such that for all i ∈ [ t ] there exists j ∈ [2] such that H i ⊆ G j . ∗ Department of Mathematics, Miami University, [email protected] . Research supported in part byNSF grant DMS-1954170. † Alfr´ed R´enyi Institute of Mathematics, Budapest, P.O. Box 127, Budapest, Hungary, H-1364. [email protected] . Research supported in part by NKFIH Grant No. K132696.
Problem 1.1.
Given a -colored complete digraph K , is it possible to cover K with at mostfour monochromatic -dominating subgraphs? (If not four, some other fixed number?) Our main result is a positive answer for the qualitative part of Problem 1.1 in a moregeneral form.
Theorem 1.2.
Let d be an integer with d ≥ . In every -colored complete digraph K , thereexists a cover of K with at most × P di =1 d i = 2 d (cid:16) d d − d − (cid:17) monochromatic d -dominatingsubgraphs. In case of d = 2 there exists a cover of K with at most eight monochromatic -dominating subgraphs. For all integers d ≥
1, let f ( d ) be the minimum number of monochromatic d -dominatingsubgraphs needed to cover an arbitrarily 2-colored complete digraph. Note that obviously f (1) = 2 since the two sets of monochromatic loops provide an optimal cover. For d ≥
2, Theorem 1.2 shows that f ( d ) is well-defined. Example 1.3 below (adapted from [4,Proposition 6.3]) combined with Theorem 1.2 gives4 ≤ f (2) ≤ d ≤ f ( d ) ≤ d (cid:18) d d − d − (cid:19) for all integers d ≥ . (1) Example 1.3.
Let K be a complete digraph on at least d vertices and partition V ( K ) intonon-empty sets R , . . . , R d and B , . . . , B d , color all edges inside R i red, all edges inside B i blue, all edges from R i to B j red, all edges from B i to R j blue, all edges between R i and R j with i = j blue, and all edges between B i and B j with i = j red. One can check thatevery monochromatic d -dominating subgraph of K is entirely contained inside one of thesets R , . . . , R d , B , . . . , B d . Finally, the following example shows that for d ≥ Example 1.4.
Let V be a totally ordered set and let K be the complete digraph on V wherefor all i ∈ V , ( i, i ) is green and for all i, j ∈ V with i < j , ( i, j ) is red and ( j, i ) is blue.Note that for d ≥ the only monochromatic d -dominating subgraphs are the green loops andthus no bound can be put on the number of monochromatic d -dominating subgraphs neededto cover V . A graph G is d -degenerate if there is an ordering of the vertices v , v , . . . such that for all i ≥ | N ( v i ) ∩ { v , . . . , v i − }| ≤ d (equivalently, every subgraph has a vertex of degree atmost d ). Burr and Erd˝os conjectured [3] that for all positive integers d , there exists c d > K n contains a monochromatic copy of every d -degenerate graphon at most c d n vertices. This conjecture was recently confirmed by Lee [8].The motivation for Problem 1.1 relates to the following conjecture also raised in [4,Problem 1.5, Conjecture 10.2] which can be thought of as an infinite analogue of the Burr-Erd˝os conjecture. Conjecture 1.5.
For all positive integers d , there exists a real number c d > such thatif G is a countably infinite d -degenerate graph with no finite dominating set, then in every -coloring of the edges of K N , there exists a monochromatic copy of G with vertex set V ⊆ N such that the upper density of V is at least c d . d = 1 was solved completely in [4] (regardless of whether G has a finite dom-inating set or not). For certain 2-colorings of K N , described below, Theorem 1.2 implies apositive solution to Conjecture 1.5 for d ≥ F ⊆ N , we have a partition of N \ F into (finitely orinfinitely many) infinite sets X = { X , . . . , X n , . . . } . Also suppose that we have ultrafilters U , U , . . . , U n , . . . on N such that for all i ≥ X i ∈ U i . Finally, suppose that for all i, j ≥ c i,j ∈ [2] such that for all v ∈ X i , { u : { u, v } has color c i,j } ∩ X j ∈ U j . This lastcondition ensures that if there exists X i , . . . , X i n and X j such that c i ,j = · · · = c i n ,j =: c ,then every finite collection of vertices in X i ∪· · ·∪ X i n has infinitely many common neighborsof color c in X j . Note that such a scenario can be realized as follows: For all i, j , let c i,j ∈ [2]and color the edges from X i to X j so that every vertex in X i is incident with cofinitely manyedges of color c i,j (by using the half graph coloring when c i,j = c j,i for instance).The above coloring of K N naturally corresponds to a 2-colored complete digraph in thefollowing way: Let K be a 2-colored complete digraph on X where we color ( X i , X j ) withcolor c if for all v ∈ X i , { u : { u, v } has color c } ∩ X j ∈ U j . Now by Theorem 1.2, K canbe covered by t ≤ f ( d + 1) monochromatic ( d + 1)-dominating subgraphs G , . . . , G t . Since N \ F = S i ∈ [ t ] (cid:16)S X ∈ V ( G i ) X (cid:17) , there exists i ∈ [ t ] such that V i := S X ∈ V ( G i ) X has upperdensity at least 1 /f ( d + 1). Without loss of generality, suppose the edges of G i are red. Bythe construction, V i has the property that for all S ⊆ V i with 1 ≤ | S | ≤ d + 1, there is aninfinite subset W ⊆ V i such that every edge in E ( S, W ) is red. As shown in [4, Proposition6.1], if G is a graph satisfying the hypotheses of Conjecture 1.5, then there exists a red copyof G which spans V i and thus has upper density at least 1 /f ( d + 1). For a graph G , we denote the order of a largest clique (pairwise adjacent vertices) in G by ω ( G ). Given a 2-colored complete digraph K and a set U ⊆ V ( K ), define G [ U ] blue to bethe graph on U where { u, v } ∈ G [ U ] blue if and only if ( u, v ) and ( v, u ) are blue in K ; define G [ U ] red analogously.Given positive integers ω and d , let f ( ω, d ) be the smallest positive integer D suchthat if K is a 2-colored complete digraph on vertex set V where every loop has the samecolor, say red, and ω ( G [ V ] blue ) = ω , then V can be covered by at most D monochromatic d -dominating subgraphs. Also define f (0 , d ) = 0. Lemma 2.1. (1) f (1 ,
2) = 1 (2) f ( ω, d ) ≤ d ( f ( ω − , d ) + 1) for all ≤ ω ≤ d (in particular, f (1 , d ) ≤ d ). In fact, all d -dominating subgraphs in the covering have the same color as the loops. Note that the upper bound ω ≤ d is not strictly necessary, but we include it here forclarity since in the next lemma, we will prove a stronger result when ω ≥ d + 1. Proof.
Let K be a 2-colored complete digraph on vertex set V where all loops have the samecolor, say red.(1) is trivial since for all distinct u, v ∈ V both ( u, u ) and ( v, v ) are red and ω ( G [ V ] blue ) =1 implies that either ( u, v ) or ( v, u ) is red.To see (2), note first that we may assume that K itself is not spanned by a red d -dominating subgraph, otherwise we are done. This is witnessed by a set U = { u , . . . , u d } ⊆ V , such that there is no w ∈ V with ( u i , w ) red for all i ∈ [ d ]. Given a totally ordered set Z and disjoint X, Y ⊆ Z the half graph coloring of the complete bipartitegraph K X,Y is a 2-coloring of the edges of K X,Y where for all i ∈ X , j ∈ Y , { i, j } is red if and only if i ≤ j . i ∈ [ d ] we define W i = { v ∈ V : ( v, u i ) is red } . Note that u i ∈ W i and K [ W i ] is spanned by a red d -dominating subgraph for all i ∈ [ d ].Set V ′ = V \ ( ∪ i ∈ [ d ] W i ) and define T i = { v ∈ V ′ : ( u i , v ) is blue } . Note, that by the definition of V ′ , ( v, u i ) is also blue for all v ∈ T i and i ∈ [ d ]. Moreover,from the selection of U , every vertex in V ′ receives a blue edge from some vertex in U andtherefore V ′ = ∪ di =1 T i .Note that if ω = 1, then T i = ∅ for all i ∈ [ d ] and thus ∪ i ∈ [ d ] W i is a cover of K with d red d -dominating subgraphs; i.e. f (1 , d ) ≤ d = d ( f (0 , d ) + 1).Otherwise, we have that ω ( K [ T i ] blue ) ≤ ω − K is covered by at most d + d · f ( ω − , d ) = d ( f ( ω − , d ) + 1)red d -dominating subgraphs. Lemma 2.2.
Let K be a 2-colored complete digraph K where R is the set of red loops and B is the set of blue loops. If ω ( G [ R ] blue ) ≥ d + 1 , then V ( K ) can be covered by at most d red d -dominating subgraphs and at most one blue d -dominating subgraph. Likewise, if ω ( G [ B ] red ) ≥ d + 1 . In particular, this implies f ( ω, d ) ≤ d + 1 for ω ≥ d + 1 .Proof. Suppose ω ( G [ R ] blue ) ≥ d + 1 and let X = { x , . . . , x d , x d +1 } ⊆ R be a set of order d + 1 which witnesses this fact. For i ∈ [ d ] we define W i = { v ∈ V ( K ) : ( v, x i ) is red } . Note that x i ∈ W i and K [ W i ] is spanned by a red d -dominating subgraph for all i ∈ [ d ].Set V ′ = X ∪ ( V ( K ) \ ( ∪ i ∈ [ d ] W i )) and note that for all v ∈ V ′ , [ v, X ] is blue. Now let S ⊆ V ′ such that 1 ≤ | S | ≤ d . If S ⊆ X , then since | S | < | X | , there exists x i ∈ X \ S suchthat every edge in [ S, x i ] is blue; otherwise | S ∩ X | ≤ d − i ∈ [ d ] such that x i / ∈ S and every edge in [ S, x i ] is blue. So there is one blue d -dominating subgraph whichcovers V ′ , which together with the red d -dominating subgraphs K [ W ] , . . . , K [ W d ] gives theresult.When ω ( G [ B ] red ) ≥ d + 1, the proof is the same by switching the colors.Now we are ready to prove our main result. Proof of Theorem 1.2.
Let V ( K ) = R ∪ B where R, B are the vertex sets of the red andblue loops, respectively. If ω ( G [ R ] blue ) ≥ d + 1 or ω ( G [ B ] red ) ≥ d + 1, then by Lemma 2.2, R ∪ B can be covered by at most d + 1 monochromatic d -dominating subgraphs. So suppose ω ( G [ R ] blue ) ≤ d and ω ( G [ B ] red ) ≤ d . Now by Lemma 2.1, each of K [ R ] and K [ B ] can becovered by at most 4 monochromatic d -dominating subgraphs when d = 2, and by at most P ωi =1 d i ≤ P di =1 d i monochromatic d -dominating subgraphs when d ≥ In the above section, we proved that f (1 ,
2) = 1 and f (1 , d ) ≤ d for all d ≥
3. Naturally,we wondered if the upper bound on f (1 , d ) could be improved when d ≥ f (1 , d ) would improve the general upper bound on f ( d )). In this sectionwe show that it cannot; that is, f (1 , d ) = d for all d ≥ tournament is a digraph ( V, E ) such that for all distinct x, y ∈ V exactly one of( x, y ) , ( y, x ) is in E and ( x, x ) / ∈ E . Given a digraph G = ( V, E ), we say that S ⊆ V isan out-dominating set if for all v ∈ V \ S , there exists u ∈ S such that ( u, v ) ∈ E , andwe say that S is an in-dominating set if for all v ∈ V \ S , there exists u ∈ S such that( v, u ) ∈ E . Note that a tournament T is d -dominating ( d -dominated) if and only if T hasno in-dominating (out-dominating) set of order d .We call a d -dominating ( d -dominated) tournament critical if its proper subtournamentsare not d -dominating ( d -dominated). For a tournament T , let T ∗ be the digraph obtainedfrom T by adding a loop at every vertex.Our main result of this section is the following. Theorem 3.1.
For all integers d ≥ , if T is a critical d -dominated tournament with no ( d + 1) -dominating subtournaments, then f (1 , d + 1) = d + 1 . However, before proving Theorem 3.1, we show that such a tournament exists for all d ≥ Corollary 3.2.
For all d ≥ , f (1 , d ) = d . Note that the absence of loops and two-way oriented edges make the existence of d -dominated tournaments a nontrivial problem. This existence problem for d -dominatedtournaments was proposed by Sch¨utte (see [5]) and was first proved by Erd˝os [5] withthe probabilistic method, then Graham and Spencer [6] gave an explicit construction usingsufficiently large Paley tournaments .Note that Babai [1] coined the term d -paradoxical tournament for what we refer to as d -dominated tournament. In this spirit, we say that a tournament is perfectly d -paradoxical if it is d -dominating, d -dominated, has no ( d + 1)-dominating subtournaments, and has no( d + 1)-dominated subtournaments. A result of Esther and George Szekeres [7] combinedwith the fact that Paley tournaments are self-complementary implies that QT is perfectly2-paradoxical and QT is perfectly 3-paradoxical. It is an open question (which to the bestof our knowledge we are raising here for the first time) whether every Paley tournamentis perfectly d -paradoxical for some d . While we can’t settle that question, the followingbeautiful example of Bukh [2] shows that perfectly d -paradoxical tournaments exist for all d ≥
2. We repeat his proof here (tailored to the terminology of this paper) for completeness.
Example 3.3 (Bukh [2]) . For all integers d ≥ , there exists a perfectly d -paradoxicaltournament. In particular, there exists a critical d -dominated tournament which has no ( d + 1) -dominating subtournaments.Proof. Let d be an integer with d ≥ n = m ( d + 1) where m = 2 d . Let V = { , , . . . , n − } and let G be the oriented graph on V where ( i, j ) ∈ E ( G ) if and only if1 ≤ j − i ≤ m − n ). In other words G is the oriented ( m − n vertices. Now we define a tournament T by starting with the oriented graph G and for all distinct i, j ∈ V , if ( i, j ) , ( j, i ) E ( G ), then independently and uniformly atrandom let ( i, j ) ∈ E ( T ) or ( j, i ) ∈ E ( T ).First note that every induced subgraph of G has an in-dominating set of order at most d + 1 and an out-dominating set of order at most d + 1 and thus the same is true of everysubtournament of T . This implies that T has no ( d + 1)-dominating subtournaments andno ( d + 1)-dominated subtournaments.Now we claim that with positive probability, T has no out-dominating sets of order d and no in-dominating sets of order d and thus T is d -dominated and d -dominating. Let For a prime power p , p ≡ − QT p is defined on vertex set V = [0 , p − a, b ) is a directed edge if and only if a − b is a non-zero square in the finite field F p . ⊆ V with | S | = d and let N + G [ S ] = { v ∈ V : v ∈ S or there exists u ∈ S such that ( u, v ) ∈ E ( G ) } . Let V ′ := V \ N + G [ S ] and note that | N + G [ S ] | ≤ dm and thus | V ′ | ≥ m . The probability that v ∈ V ′ is dominated by S in T is 1 − − d and thus the probability that every vertex in V ′ isdominated by S is (1 − − d ) | V ′ | ≤ (1 − − d ) m ≤ e − − d m = e − d . Likewise for every vertexof V ′ dominating S . So the expected number of out-dominating or in-dominating sets oforder d is at most 2 (cid:18) nd (cid:19) e − d < em ) d e − d < e d +1 ) d e − d < d + 1) d < d for all d ≥ d -paradoxical tournament T , let T ′ be a minimal subtournamentof T which is d -dominated. So T ′ is critical d -dominated and has no ( d + 1)-dominatingsubtournaments.The proof of Theorem 3.1 will follow from two more general lemmas. Lemma 3.4.
Let T be a tournament and let d ≥ . If T is 2-dominating and there existsa set W ⊆ V ( T ) with | W | = d such that W dominates exactly one vertex v , then T ∗ isnot ( d + 1) -dominating. In particular, if T is critical d -dominating, then T ∗ is not ( d + 1) -dominating.Proof. Let W = { w , . . . , w d } and v be as in the statement. To see that T ∗ is not ( d + 1)-dominating, it is enough to prove that for some u ∈ N + ( v ) the set W ∪ { u } does notdominate any vertex in T ∗ (note that since T is 2-dominating, N + ( v ) = ∅ ). Suppose forcontradiction that this is not the case; that is, for all u ∈ N + ( v ) the set W ∪ { u } dominatessome vertex x in T ∗ . Note that by the definition of W and the fact that u ∈ N + ( v ), it mustbe the case that x ∈ W ; without loss of generality, suppose x = w . This implies that for all i ∈ [ d ], ( w i , w ) ∈ E ( T ). But now this implies that for all u ∈ N + ( v ), W ∪ { u } dominates w . On the other hand since T is 2-dominating, it must be the case that there exists avertex which is dominated by { w , v } in T , but every outneighbor of v is an inneighbor of w and thus we have a contradiction.To get the second part of the lemma, first note that if T is critical d -dominating, then T is 2-dominating. Moreover, for all v ∈ V , since T − v is not d -dominating there exists W = { w , . . . w d } ⊆ V ( T ) \ { v } which does not dominate any vertex in V ( T ) \ { v } , but since T is d -dominating, W must dominate v .If G = ( V, E ) is a digraph such that there exists w ∈ V such that ( v, w ) ∈ E for all v ∈ V (including v = w ), then note that G is d -dominating for all d ≤ | V | . In this case wecall G trivially d -dominating . Lemma 3.5.
Let T be a tournament. If T is critical d -dominating, then T ∗ cannot becovered by less than d + 1 ( d + 1) -dominating subgraphs.Proof. Suppose for contradiction that for some t ≤ d there are ( d +1)-dominating subgraphs H , . . . , H t which cover T ∗ . Since T is critical d -dominating we have by Lemma 3.4 that T ∗ is not ( d + 1)-dominating, and thus all V ( H i ) are proper subsets of V ( T ∗ ). Claim 3.6.
Each H i is trivially ( d + 1) -dominating. roof. The claim is obvious if | V ( H i ) | ≤ d ; so suppose that | V ( H i ) | ≥ d + 1. Since T iscritical d -dominating, the subtournament T i of T spanned by V ( H i ) is not d -dominating.This is witnessed by a set W = { w , . . . , w d } ⊆ V ( T i ) such that W does not dominate anyvertex in U = V ( T i ) \ W . Let u ∈ U . Since H i is ( d + 1)-dominating, W ∪ u dominates somevertex x ∈ V ( H i ) which must be in W from the definition of W . Without loss of generality,let x = w . This implies that ( u, x ) ∈ E ( T ) and for all i ∈ [ d ], ( w i , w ) ∈ E ( T ). Butnow this implies that for all u ∈ U , W ∪ { u } dominates w and thus all vertices of V ( H i )(including w ) are oriented to w proving the claim.Claim 3.6 implies that for all i ∈ [ t ] there is a vertex v i ∈ V ( H i ) which is dominated byall vertices of H i . But since ∪ ti =1 V ( H i ) = V ( T ), the set { v , . . . , v t } does not dominate anyvertex in T , contradicting the fact that T is d -dominating. Proof of Theorem 3.1.
First note that f (1 , d + 1) ≤ d + 1 by Lemma 2.1.Let T B be a tournament on vertex set V such that T B is critical d -dominated and hasno ( d + 1)-dominating subtournaments. Define the 2-colored complete digraph K on V bycoloring all edges of T B blue, and all edges of ( V × V ) \ E ( T B ) red. Let T R be the tournamentwith E ( T R ) = { ( y, x ) : ( x, y ) ∈ E ( T B ) } and note that every edge of T R is red and T R has noloops. Since T B is critical d -dominated, this implies that T R is critical d -dominating (since T R is obtained by reversing all the edges of T B ).Note that by the assumption on T B , every monochromatic ( d + 1)-dominating subgraphin K must be red. However, since T R is crtical d -dominating, we get that f (1 , d + 1) ≥ d + 1from Lemma 3.5. Acknowledgements.
We thank Boris Bukh for Example 3.3 and for his comments onthe paper.
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