Polychromatic colorings of complete graphs with respect to 1-,2-factors and Hamiltonian cycles
Maria Axenovich, John Goldwasser, Ryan Hansen, Bernard Lidický, Ryan R. Martin, David Offner, John Talbot, Michael Young
aa r X i v : . [ m a t h . C O ] D ec Polychromatic colorings of complete graphs with respectto 1-, 2-factors and Hamiltonian cycles
Maria Axenovich ∗ John Goldwasser † Ryan Hansen ‡ Bernard Lidick´y § Ryan R. Martin ¶ David Offner k John Talbot ∗∗ Michael Young †† August 12, 2018
Abstract If G is a graph and H is a set of subgraphs of G , then an edge-coloring of G is called H -polychromatic if every graph from H gets all colors present in G on its edges. The H -polychromatic number of G , denoted poly H ( G ), is the largest number of colors inan H -polychromatic coloring. In this paper, poly H ( G ) is determined exactly when G is a complete graph and H is the family of all 1-factors. In addition poly H ( G ) is foundup to an additive constant term when G is a complete graph and H is the family of all2-factors, or the family of all Hamiltonian cycles. If G is a graph and H is a set of subgraphs of G , we say that an edge-coloring of G is H - polychromatic if every graph from H gets all colors present in G on its edges. The H -polychromatic number of G , denoted poly H ( G ), is the largest number of colors in an H -polychromatic coloring. If an H -polychromatic coloring of G uses poly H ( G ) colors, it iscalled an optimal H -polychromatic coloring of G . ∗ Karlsruhe Institute of Technology, Karlsruhe, Germany, [email protected] . † West Virginia University, Morgantown, WV, USA, [email protected] . ‡ West Virginia University, Morgantown, WV, USA, [email protected] § Iowa State University, Ames, IA, USA, lidicky@iastate,edu . Supported by NSF grant DMS-1600390. ¶ Iowa State University, Ames, IA, USA, [email protected] . Research supported in part by SimonsFoundation Collaboration Grant ( k Westminster College, New Wilmington, PA, USA, [email protected] . ∗∗ University College London, London, UK, [email protected] . †† Iowa State University, Ames, IA, USA, [email protected] . .1 Background Let Q n denote the hypercube of dimension n . Let G = Q n and H be the family of allsubgraphs of G isomorphic to Q d . If d is fixed and n is large, then Alon, Krech, andSzab´o [3] showed that ⌊ ( d +1) ⌋ ≤ poly H ( Q n ) ≤ (cid:0) d +12 (cid:1) . Offner [11] proved that the lowerbound is tight for all sufficiently large values of n . Bialostocki [4] treated the special casewhen d = 2 and n ≥
2. Goldwasser et al. [9] considered the case where H is the family ofall subgraphs of Q n isomorphic to a Q d minus an edge or a Q d minus a vertex.If T is a tree and H is the set of all paths of length at least r , then poly H ( T ) = ⌈ r/ ⌉ ,as was shown by Bollob´as et al. [5]. When G = K n and H is the set of all r -vertex cliques,poly H ( G ) was considered by Erd˝os and Gy´arf´as [6, 10] with the respective colorings calledbalanced. When G is an arbitrary multigraph of minimum degree d , and H is the set of allstars with center v and leaves N ( v ), v ∈ V ( G ), then it was shown by Alon et al. [2], thatpoly H ( G ) ≥ ⌊ (3 d + 1) / ⌋ . Goddard and Henning [7] considered vertex-colorings of graphssuch that each open neighborhood contains a vertex of every color used in G .Polychromatic colorings were also investigated for vertex-colored hypergraphs. Thesecolorings are essential tools in studying covering problems which are of fundamental impor-tance in general graph and hypergraph settings, especially in geometric hypergraphs, andthey exhibit connections to VC-dimension, see [1, 2, 5, 12]. In this paper, we consider the case where G is a complete graph and H is a family of spanningsubgraphs. Let F = F ( n ) be the family of all 1-factors of K n , F = F ( n ) be the familyof all 2-factors of K n and HC = HC( n ) be the family of all Hamiltonian cycles of K n . Ourmain results are as follows: Theorem 1. If n is an even positive integer, then poly F ( K n ) = ⌊ log n ⌋ . Theorem 2.
There exists a constant c such that ⌊ log n + 1) ⌋ ≤ poly F ( K n ) ≤ poly HC ( K n ) ≤⌊ log n ⌋ + c . Moreover, j log n − k ≤ poly HC ( K n ) . It is claimed in a follow-up paper [8], that in fact poly F ( K n ) = ⌊ log n + 1) ⌋ andpoly HC ( K n ) = j log n − k for n ≥
3. However, the arguments there include more caseanalysis and greater detail than what is required for the small additive constant given inTheorem 2.The paper is structured as follows. We start with basic definitions in Section 2. InSection 3, we give constructions of polychromatic colorings, which provide the lower boundsfor Theorems 1 and 2. In Section 4, we prove Theorem 1. Section 5 contains the proof ofTheorem 2. 2
Definitions
Let the vertices of K n be denoted by v , v , . . . , v n . An edge-coloring ϕ is ordered at v i for i ∈ [ n ] if there exists a color a , called the main color at v i , such that ϕ ( v i v j ) = a for all j ∈ { i + 1 , . . . , n } . Notice that v n − and v n are ordered with respect to any coloring. Wedefine the main color of v n to be the same as for v n − . A vertex v i is unitary if there arecolors a = b such that v i is incident with n − a and one edge v i v j colored b , where v j is unitary with n − b . For v i unitary, we also call a the main color .An edge-coloring is ordered if all vertices are ordered with respect to some ordering ofvertices. See Figure 1 for an example of an ordered coloring. We call an edge-coloring combed if each vertex is either ordered or unitary. It is not difficult to show that if there is at leastone unitary vertex in a combed coloring then either the first three vertices (and no others)are unitary with different main colors, as in Figure 2, or the first four vertices (and no others)are unitary with two of them with one main color, and two with another.Let ϕ be an ordered or combed coloring. The inherited coloring is the vertex-coloring ϕ ′ obtained by coloring each vertex with its main color. Its inherited color class M i of color i is the set of all vertices v with ϕ ′ ( v ) = i . Let M t ( j ) = M t ∩ { v , v , . . . , v j } . In this paper,we shall always think of the ordered vertices as arranged on a horizontal line with v i to theleft of v j if i < j . We say that an edge v i v j , i < j , goes from v i to the right and from v j tothe left. If ϕ is an edge-coloring of a graph G , the maximum monochromatic degree of G isthe largest integer d such that some vertex of G is incident to d edges of the same color. Wesay such a vertex is a max-vertex . If X is a subset of V ( K n ), we say that the edge-coloring ϕ of K n is • X -constant if for any v ∈ X , ϕ ( vu ) = ϕ ( vw ) for all u, w ∈ V \ X , • X -ordered if there is an ordering of the vertices such that X = { v , . . . , v m } for someinteger m and ϕ is ordered on vertices in X .Notice if a coloring ϕ is X -ordered, it is also X -constant. We construct three edge-colorings of K n , and show that they are polychromatic for F , F ,and HC, respectively. F -polychromatic Coloring ϕ F Let n ≥ k be the largest positive integer such that 2 k ≤ n , i.e., k = ⌊ log n ⌋ .Let ϕ ′ be a vertex-coloring of K n with vertex set { v , . . . , v n } and colors 1 , . . . , k , where for3 v v v v v v v v v · · · · · · · · · M M M M Figure 1: F -polychromatic coloring ϕ F .each i ∈ [ k ], M i is the color class of color i . Moreover, for any 1 ≤ i < j ≤ k , every vertexin M i precedes every vertex in M j , and | M t | = 2 t − for t = 1 , . . . , k −
1. Hence the colorclasses 1 , , . . . , k have sizes 1 , , , . . . , k − , n − k − + 1, respectively. Let ϕ F be the orderedcoloring for which ϕ ′ is the inherited coloring.Consider an arbitrary 1-factor F of K n and t ∈ [ k ]. Consider the set F t of all edgesof F with at least one endpoint in M t . Since | M | + · · · + | M i | = 2 i − | M k | = n − | M | − · · · − | M k − | ≥ k − k − + 1 = 2 k − + 1, we have P i 1, i.e., k = 1+ ⌊ log ( n +1) ⌋ . Let ϕ ′ bea vertex-coloring of K n with vertex set { v , . . . , v n } and colors 1 , . . . , k , where for each i ∈ [ k ], M i is the color class of color i . Moreover, for any 1 ≤ i < j ≤ k , every vertex in M i precedesevery vertex in M j , and | M t | = 2 t − for t = 4 , . . . , k − 1, and | M | = | M | = | M | = 1. Hencethe color classes 1 , , . . . , k − , k have sizes 1 , , , , , . . . , k − , n − k − + 1, respectively.Let ϕ F be obtained by taking the ordered coloring for which ϕ ′ is the inherited coloring andthen recoloring the edge v v from color 1 to color 3. See Figure 2.Observe that the inherited color classes M , M , and M contain unitary vertices. More-over, | M | + · · · + | M t | = 2 t − − ≤ t ≤ k − 1, and | M k | = n − | M | − · · · − | M k − | ≥ k − − − k − + 1 = 2 k − . So, | M t | > P i 4. Consider an arbitrary2-factor F of K n and t ∈ [ k ]. For i ≤ v i is a unitary vertex with main color i , so F musthave edges of colors 1, 2, and 3. For a color t ≥ 4, consider the set F t of edges of F withendpoints in M t . Then F t has an edge of color t unless F t forms a bipartite graph G t withone part M t and another M ′ t = S t − i =1 M i . The degree of each vertex of G t from M t is two, andthe degree of each vertex of G t from M ′ t is at most two. Thus | M ′ t | ≥ | M t | , a contradiction.Thus, F t , and therefore F , has at least one edge of color t . So, ϕ F is F -polychromatic and4 v v v v v v v v v · · · · · · · · · M M M M M Figure 2: F -polychromatic coloring ϕ F . v v v v v v v v v v v v · · · M M M M M Figure 3: HC-polychromatic coloring ϕ HC .it uses k = ⌊ log n + 1) ⌋ colors. HC -polychromatic Coloring ϕ HC Let k be the largest positive integer such that n ≥ · k − +1, i.e., k = 3+ ⌊ log ( n − / ⌋ . Let ϕ ′ be a vertex-coloring of K n with vertex set { v , . . . , v n } and colors 1 , . . . , k , where for each i ∈ [ k ], M i is the color class of color i . Moreover, for any 1 ≤ i < j ≤ k , every vertex in M i precedes every vertex in M j , and | M t | = 3 · t − for t = 4 , . . . , k − 1, and | M | = | M | = | M | =1. Hence the color classes 1 , , . . . , k − , k have sizes 1 , , , , , , . . . , · k − , n − · k − ,respectively. Let ϕ HC be obtained by taking the ordered coloring for which ϕ ′ is the inheritedcoloring and then recoloring the edge v v from color 1 to color 3. See Figure 3.We have that | M | + · · · + | M t | = 3 · t − for 3 ≤ t ≤ k − 1. Moreover, | M k | = n − | M | − · · · − | M k − | ≥ · k − + 1 − · k − = 3 · k − + 1 . Thus | M k | > P i 4. Consider an arbitrary Hamiltonian cycle H of K n . For i ≤ v i is a unitary vertex with main color i , so H must have edges of colors 1, 2, and3. For each color t ≥ 4, let H t be the set of edges of H with at least one endpoint in M t .5hen H t has an edge of color t unless H t forms a bipartite graph G t with one part M t andanother M ′ t = S t − i =1 M i . The degree of each vertex of G t from M t is two, and the degree ofeach vertex of G t from M ′ t is at most two. If 4 ≤ t < k , | M t | = | M ′ t | , the degree of eachvertex of G t from M ′ t is also two. Hence G t is a union of cycles, so it could not be a propersubgraph of a Hamiltonian cycle. If t = k , | M k | > | M ′ k | , so a bipartite graph G t could notexist. Thus H has an edge of color t for each t = 1 , . . . , k , ϕ HC is HC-polychromatic, and ituses j log n − k colors. We prove Theorem 1 by first showing the existence of an optimal edge-coloring that isordered. Then we use Lemma 3 below which states that, for every inherited color class M t ,there exists j such that a majority of v , . . . , v j is in M t . This leads to a counting argumentthat gives the upper bound in Theorem 1. For the lower bound we use the coloring ϕ F . Lemma 3. Let ϕ : E ( K n ) → [ k ] , where n is even, be an ordered coloring with inheritedcolor classes M , . . . , M k . If the coloring ϕ is F -polychromatic, then ∀ t ∈ [ k ] ∃ j ∈ [ n − such that | M t ( j ) | > j/ .Proof. Assume there exists t such that for each j ∈ [ n − | M t ( j ) | ≤ j/ 2. Let x , . . . , x m be the vertices of M t in order and let y , . . . , y n − m be the other vertices of K n in order. Let H consist of the edges y x , y x , . . . , y m x m and a perfect matching on { y m +1 , . . . , y n − m } (ifthis set is non-empty). Since | M t ( j ) | ≤ j/ j , the number of y ’s that must precede x i is at least i for each i = 1 , . . . , m . Hence y i is to the left of x i for each i = 1 , . . . , m .Therefore all edges in H incident with vertices in M t go to the left and do not have color t .The edges of H that are not incident with vertices in M t are also not of color t . Hence ϕ isnot F -polychromatic, a contradiction. Proof of Theorem 1. Let k = poly F ( K n ) be the polychromatic number for 1-factors in K n =( V, E ). Among all F -polychromatic colorings of K n with k colors we choose ones that are X -ordered for a subset X (possibly empty) of the largest size, and, of these, choose a coloring ϕ whose restriction to V \ X has the largest maximum monochromatic degree. Suppose forcontradiction that V = X .Let Z = V \ X and G be the subgraph of K n induced by Z . Let v be a vertex of maximummonochromatic degree, d , in ϕ restricted to G , and let 1 be a color for which there are d edges incident with v in G with color 1. By the maximality of | X | , there is a vertex u in Z such that ϕ ( uv ) = 1. Assume ϕ ( uv ) = 2. If every 1-factor containing uv had anotheredge of color 2, then the color of uv could be changed to 1, resulting in an F -polychromaticcoloring where v has a larger maximum monochromatic degree in G , a contradiction. Hence,there is a 1-factor F in which uv is the only edge with color 2 in ϕ .6et ϕ ( vy i ) = 1, y i ∈ Z , i = 1 , . . . , d . For each i ∈ [ d ], let y i w i be the edge of F containing y i (perhaps w i = y j for some j = i ); see Figure 4. We can get a different 1-factor F i byreplacing the edges uv and y i w i in F with edges vy i and uw i . Since F i must have an edge ofcolor 2 and ϕ ( vy i ) = 1, we must have ϕ ( uw i ) = 2 for each i ∈ [ d ]. v uy y y = w y = w . . . y d w w . . . w d 21 1 1 1 1Figure 4: Maximum polychromatic degree in an F -polychromatic coloring.If w i ∈ X for some i then, since ϕ is X -constant, ϕ ( w i y i ) = ϕ ( w i u ) = 2, so y i w i and uv are two edges of color 2 in F , a contradiction. So, w i ∈ Z for all i ∈ [ d ]. Thus ϕ ( uv ) = ϕ ( uw ) = · · · = ϕ ( uw d ) = 2, and the monochromatic degree of u in G is at least d + 1, larger than that of v , a contradiction.We conclude that X = V . Hence ϕ is an ordered F -polychromatic coloring of K n . ByLemma 3, for every t ∈ [ k ] there exists j t such that | M t ( j t ) | > j t . By permuting the colors,we can assume j t < j t whenever t < t . This gives us an ordering of inherited colorclasses M , M , . . . , M k . Since | M | ≥ | M t ( j t ) | > j t , we can use induction to show | M t | ≥ | M t ( j t ) | ≥ t − as follows | M t | ≥ | M t ( j t ) | > X ≤ i Recall that we call an edge-coloring ϕ combed if all vertices are either ordered or unitary.We prove Theorem 2 by first showing the existence of an optimal edge-coloring that iscombed. Then we use Lemma 4 below which states that, for every inherited color class M t ,either there exists j such that at least half of v , . . . , v j is in M t or M t contains a unitaryvertex. This leads to a counting argument that finishes the proof of Theorem 2. Lemma 4. Let ϕ : E ( K n ) → [ k ] be a combed coloring with inherited color classes M , . . . , M k .If the coloring ϕ is F -polychromatic, or HC -polychromatic, then ∀ t ∈ [ k ] ∃ j ∈ [ n − suchthat | M t ( j ) | ≥ j or M t contains a unitary vertex.Proof. Let H ∈ { F , HC } . Let ϕ be a combed H –polychromatic coloring with inherited colorclasses M , . . . , M k . It is sufficient to consider an arbitrary color t ∈ [ k ] and show that thecondition on M t is satisfied.Let x , . . . , x m be the vertices of M t in order and let y , . . . , y n − m be the other verticesof K n in order. Suppose for contradiction that there exists t such that | M t ( j ) | < j forall j ∈ [ n − 1] and M t does not contain a unitary vertex. Thus ϕ is ordered at each x i ∈ M t and so y i +1 is to the left of x i for each i ∈ [ m ]. Consider a Hamiltonian cycle H = y x y x · · · y m x m y m +1 · · · y n − m y .Since | M t ( j ) | < j/ j , the number of y ’s that must precede x i is at least i + 1 foreach i = 1 , . . . , m . Hence y i and y i +1 are to the left of x i for each i = 1 , . . . , m . Thereforeeach edge in H incident with a vertex x i in M t goes to the left from the perspective of x i .Let yx be an edge of H , where x ∈ M t . Since y M t , the majority color r of y is not t . Since ϕ is combed, either ϕ ( yx ) = r or ϕ ( yx ) = r and both y and x are unitary vertices.Recall M t does not contain any unitary vertices. Hence no edge in H is colored by t . Thiscontradicts the fact that ϕ is H -polychromatic.We say that a Hamiltonian cycle H ′ is obtained from a Hamiltonian cycle H by a twist of disjoint edges e and e of H if E ( H ) \ { e , e } ⊆ E ( H ′ ), i.e. we remove e , e from H and introduce two new edges to make the resulting graph a Hamiltonian cycle. Note thatthe choice of these two edges to add is unique. The other choice of two edges to add doesnot preserve connectivity. Without the connectivity requirement, the operation is known asa .Notice that any 2-switch could be applied to a 2-factor and the result will be again a2-factor. Here, it might be possible to add the two new edges in two different ways.For H ∈ { HC , F } , Lemma 5 can be used to show that there exists an optimal H -polychromatic that is combed. 8 emma 5. Suppose n ≥ and X ⊂ V ( K n ) . Let H ∈ { HC , F } and ϕ be an optimal H -polychromatic coloring of K n that is X -constant. Then there exists an optimal H poly-chromatic coloring ϕ of K n that agrees with ϕ on all edges with at least one endpoint in X such that(A) there exists a vertex v ∈ V ( K n ) \ X such that ϕ is ( X ∪ { v } ) -constant; or(B) X = ∅ and there exist vertices x, y, z , such that these vertices are unitary under ϕ ofdistinct main colors. This implies ϕ is { x, y, z } -constant and xyz is a rainbow triangle.Proof. Let H ∈ { F , HC } . Let Z = V ( K n ) \ X and G be the subgraph of K n induced by Z .Let | Z | = m . If m ≤ X = ∅ and (A) is trivially satisfied. Hence m ≥ 3. If X = ∅ andthere exists an optimal H -polychromatic coloring ϕ with three unitary vertices x , y , and z of distinct main colors, then (B) is satisfied. Hence we assume there is no such edge-coloring ϕ . Choose ϕ to be an optimal H -polychromatic coloring such that it agrees with ϕ onedges with at least one endpoint in X and subject to this, it maximizes the maximummonochromatic degree of G . Define d to be the maximum monochromatic degree of verticesin G with ϕ .First suppose d = m − 1. Let v be a vertex of maximum monochromatic degree d in G .Then ϕ is ( X ∪ { v } )-constant and we have (A). Hence we assume d ≤ m − ℓ = m − − d and let ϕ use colors 1 , . . . , k . If color a appears d times in G at avertex v ∈ Z , we say v is an a -max-vertex . If the ℓ edges incident with v in G which do nothave color a all have color b , we call v an ( a, b ) -max-vertex with minority color b . Claim 1. If a, b ∈ [ k ] are two distinct colors, v ∈ Z is an a -max-vertex and ϕ ( vu ) = b forsome other vertex u ∈ Z , then all of the following hold:(1) u is a b -max-vertex,(2) v is an ( a, b )-max-vertex,(3) either X = ∅ or H = F , and(4) at least half of the edges between X and Z have color b . Proof. For ease of notation, we assume that a = 1 and b = 2. Let v be a 1-max-vertex. Let u ∈ Z be a vertex such that ϕ ( vu ) = 2. If every H ∈ H containing uv contains another edgeof color 2, we could change the color of uv to color 1, giving an H -polychromatic coloringwhere v has monochromatic degree d + 1, a contradiction. Hence, there must be H ∈ H where uv is the only edge of color 2.Cyclically orient the edges of each cycle in H such that uv is an arc, and denote theresulting directed graph H . Let ϕ ( vy j ) = 1, for y j ∈ Z , j = 1 , , . . . , d . For each j ∈ [ d ],9 yxqp v = w uy y y w w ... ... zw Z Figure 5: Situation in Claim 1.let w j be the predecessor of y j in H , so w j y j ∈ H for each j . We assume w j = v for j = 2 , , . . . , d , but perhaps w = v and perhaps w j = y i for some i = j . See Figure 5.Now we shall prove (1). If w j = v , twist the edges uv and w j y j of H to get a new H j ∈ H containing vy j and uw j . Since H j must have an edge of color 2 and ϕ ( vy j ) = 1,we must have ϕ ( uw j ) = 2. Hence, ϕ ( uv ) = ϕ ( uw ) = ϕ ( uw ) = · · · = ϕ ( uw d ) = 2. Notethat w j ∈ Z for each j ∈ [ d ]. This is because if w j ∈ X , then, since ϕ is X -constant and y j ∈ Z , ϕ ( w j y j ) = ϕ ( w j u ) = 2, so w j y j is another edge in H with color 2, a contradiction.That gives us d edges of color 2 at u in G . Note that if w = v , then uw is another edgeof color 2 incident to u , so w = v and vy is an arc of H . Therefore, u is a 2-max-vertex.This proves (1).Next, we prove (2), i.e., that v is a (1 , z ∈ Z be a vertex distinctfrom u such that ϕ ( vz ) = 1. Let w be the vertex such that wz is an arc in H . We knowthat w 6∈ { v = w , w , . . . , w d , u } , since z 6∈ { y , . . . , y d , v } . Let H z ∈ H , containing vz and uw , be obtained from H by twisting uv and wz . Since uv was the unique edge of H coloredby 2, either ϕ ( uw ) = 2 or ϕ ( vz ) = 2. Suppose w ∈ Z . Since the maximum monochromaticdegree is d and ϕ ( uw j ) = 2 for all j ∈ [ d ], ϕ ( uw ) = 2, so ϕ ( vz ) = 2. Suppose w ∈ X .Since ϕ ( wz ) = 2 and ϕ is X -constant, ϕ ( wz ) = ϕ ( uw ) = 2, so ϕ ( vz ) = 2. In both cases, ϕ ( vz ) = 2. Therefore, v is a (1 , X = ∅ then both (3) and (4) hold. So, assume that X = ∅ . Let H ∈ H . Assume thatthere in an edge of H with one endpoint in X and another in Z . Then there exist x ∈ X and y ∈ Z such that yx is an arc in H . We know y 6∈ { v = w , . . . , w d , u } , because thesuccessor of y in H is in X . If we twist yx and uv we get H x ∈ H containing uy and vx ,where one of these edges must have color 2. However, since ϕ ( xv ) = ϕ ( xy ) = 2, we must10ave ϕ ( yu ) = 2, and u has monochromatic degree d + 1 in G , a contradiction. Hence thereis no edge in H with one endpoint in X and another in Z , and thus X induces a 2-factor in H . In particular, since Z = ∅ , H is not a Hamiltonian cycle, and H = F . Let p, q ∈ X with pq ∈ E ( H ). Since both ( H \ { uv, pq } ) ∪ { pv, qu } and ( H \ { uv, pq } ) ∪ { pu, qv } are 2-factorsin H , and ϕ is X -constant, either ϕ ( pv ) = ϕ ( pu ) = 2 or ϕ ( qv ) = ϕ ( qu ) = 2. In fact, since ϕ is X -constant, for each edge pq of H , where p, q ∈ X , all the edges from either p or q into Z have color 2. Since H [ X ] is a union of cycles, at least half the edges between X and Z havecolor 2. This proves (3) and (4) and finishes the proof of Claim 1. Claim 2. The graph G does not contain a (1 , , , Proof. Let x , y , and z be a (1 , , , { v, u } = { x, y } , then { v, u } = { y, z } , and then with { v, u } = { z, x } , the conclusion (4) gives that at least half of the edges between X and Z have color 2, 3, and 1, respectively. Since colors 1, 2, and 3 are distinct, we conclude X = ∅ .Let H ∈ H . Observe that x , y , and z could be incident only with edges of H with colors in { , , } in ϕ , so all other colors in H come from edges not incident with these vertices.Let ϕ ⋆ be obtained from ϕ by the following modification c ⋆ ( uv ) = u = x and v = y, u = y and v = z, u = z and v = x,ϕ ( uv ) otherwise . Observe that the union of edges of H with at least one endpoint in { x, y, z } contains allcolors { , , } in ϕ ⋆ . Hence H is polychromatic in ϕ ⋆ and ϕ ⋆ is H -polychromatic. Moreover, ϕ ⋆ is { x, y, z } -constant and all the other properties of (B) hold, which is a contradiction.This finishes the proof of Claim 2. Claim 3. If v is a (1 , u ∈ Z such that ϕ ( uv ) = 2, then u is a (2 , Proof. Let v be a (1 , u ∈ Z such that ϕ ( uv ) = 2. Claim 1 implies that u is a (2 , ⋆ )-max-vertex. Suppose for contradiction that u is a (2 , v colored 2 is the same as the number of edges incident to u colored 3 and ϕ ( uv ) = 2, there is a vertex x such that ϕ ( ux ) = 3 and ϕ ( vx ) = 1. Again byClaim 1, x is a (3 , Claim 4. If there is a (1 , , b )-max-vertex for any b = 2.11 x z vs s S u = t t TW Z Figure 6: Final part of proof of Lemma 5 with only (1 , , Proof. By symmetry suppose for contradiction that v ∈ Z is a (1 , u ∈ Z is a (1 , x ∈ Z be a vertex with ϕ ( vx ) = 2. By Claim 3, x is a (2 , ϕ ( ux ) ∈ { , } ∩ { , } = { } . Now Claim 3 applied to x and u gives that u is a (1 , a, b )-max-vertex, then { a, b } = { , } .Let S be the set of all (1 , T be the set of all (2 , W = Z \ ( S ∪ T ). By Claim 3, both S and T are not empty. Edges within S and from S to W must have color 1 (because any minority color edge at a max-vertex is incident toa max-vertex of that color), edges within T and from T to W must have color 2, and eachedge between S and T must have color 1 or 2.Suppose X = ∅ . Let s ∈ S and t ∈ T . Let ϕ ⋆ be obtained from ϕ by recoloring all edgesincident to s to 1 and by recoloring all edges incident to t and not incident to s to 2. Noticethat ϕ ⋆ is H -polychromatic. This contradicts the maximality of the monochromatic degreein ϕ . Therefore, X = ∅ .By symmetry, we can assume | S | ≤ | T | . Let v ∈ S and u ∈ T be such that ϕ ( vu ) = 2.By the maximality of the monochromatic degree of v in Z , there exists H ∈ H , where uv isthe unique edge colored by 2. Recall ℓ = m − − d . Let s , . . . , s ℓ ∈ S , where ϕ ( us i ) = 1 forall i ∈ [ ℓ ]. Let H be directed cycle(s) obtained by orienting edges of H such that uv ∈ H .Let t i be a vertex such that s i t i ∈ H for all i ∈ [ ℓ ]. See Figure 6.If there is an i ∈ [ ℓ ] such that ϕ ( vt i ) = 2, then twist of of vu and s i t i contains vt i and us i and leaves no edge colored 2, which is a contradiction with ϕ being H -polychromatic.Hence ϕ ( vt i ) = 2 and t i ∈ T for all i ∈ [ ℓ ]. Notice v has exactly ℓ incident edges colored 2and the other ends of these edges must be t , . . . , t ℓ . By symmetry, we assume t = u .Suppose there exists xz in H with x ∈ X and z ∈ Z . Since uv is the unique edge of12 „ H colored 2, ϕ ( xz ) is not 2 and since ϕ is X -constant, ϕ ( xz ) = ϕ ( xu ) = 2. Notice that z 6∈ { u = t , . . . , t ℓ } since for every i ∈ [ ℓ ], the predecessor of t i in H is s i and s i ∈ S . Hence ϕ ( vz ) = 1 and the twist of xz and uv contains xu and vz . Since ϕ ( xu ) = 2 and ϕ ( vz ) = 2,we get a contradiction to ϕ being H -polychromatic. Therefore, there is no edge of H between X and Z .Since there is no edge of H between X and Z and X = ∅ , H is not connected. Therefore, H = F .Recall that all edges between T and W have color 2, hence they are not in H . Sincethere are no edges of H between X and Z , and all edges within T have color 2, every vertexin T has both neighbors from H in S . On the other hand, every vertex in S has at most twoneighbors from H in T . Thus | S | ≥ | T | . Recall we assumed | S | ≤ | T | . Hence | S | = | T | andthere are no edges of H between S ∪ T and W .Consider a bipartite graph B with vertex set S ∪ T , edges st , s ∈ S , t ∈ T , and ϕ ( st ) = 1.Since vertices in T are (2 , ℓ in B . Similarly,since vertices in S are (1 , ℓ verticesof T . Therefore, all vertices in S have the same degree in B . Since | S | = | T | , we conclude B is an ℓ -regular graph.If ℓ ≥ K in B . Let H ⋆ be obtained from H by removingedges incident to vertices in S ∪ T and adding K . Since all edges of K have color 1 and uv was the unique edge of H colored 2, we conclude H ⋆ has no edge colored 2, contradictingthe assumption that ϕ is H -polychromatic.Finally, if ℓ = 1, then B is a matching on 4 vertices and the other two edges between S and T must have color 2. Hence S ∪ T does not contain a 2-factor in which uv would be theunique edge colored 2. This contradicts the existence of H .This finishes the proof of Lemma 5. Proof of Theorem 2. Let H ∈ { F , HC } . Let ϕ be an optimal H -polychromatic coloring of E ( K n ) with k colors and [ k ] be the set of colors. We choose X = ∅ , then we repeatedly applyLemma 5. In the first application, we may get Lemma 5(B) and get X = { x, y, z } that areunitary of distinct colors or Lemma 5(A) and | X | = 1. But after that Lemma 5(A) alwaysapplies. Note that there are no unitary vertices except possibly x, y , and z because eachother vertex is incident to distinct colors c x , c y , c z that are main colors of x, y , and z . Thisresults in a combed edge-coloring ϕ with zero or three first unitary vertices and all othersbeing ordered vertices.Let M i be the inherited color classes obtained from ϕ . Let M , . . . , M k − be the inheritedcolor classes not containing x, y , or z . By Lemma 4, for each color class M t there is j t suchthat | M t ( j t ) | ≥ j t . By symmetry, assume j i < j t for all 1 ≤ i < t ≤ k − 3. This leads to | M t ( j t ) | ≥ X i 1. Hence by induction we get | M t ( j t ) | ≥ X ≤ i References [1] E. Ackerman, B. Keszegh, and M. Vizer. Coloring Points with Respect to Squares.In S. Fekete and A. Lubiw, editors, , volume 51 of Leibniz International Proceedings in Informatics(LIPIcs) , pages 5:1–5:16, Dagstuhl, Germany, 2016. Schloss Dagstuhl–Leibniz-Zentrumfuer Informatik.[2] N. Alon, R. Berke, K. Buchin, M. Buchin, P. Csorba, S. Shannigrahi, B. Speckmann,and P. Zumstein. Polychromatic colorings of plane graphs. Discrete Comput. Geom. ,42(3):421–442, 2009.[3] N. Alon, A. Krech, and T. Szab´o. Tur´an’s theorem in the hypercube. SIAM J. DiscreteMath. , 21(1):66–72, 2007. 144] A. Bialostocki. Some Ramsey type results regarding the graph of the n -cube. ArsCombin. , 16(A):39–48, 1983.[5] B. Bollob´as, D. Pritchard, T. Rothvoss, and A. Scott. Cover-decomposition and poly-chromatic numbers. SIAM J. Discrete Math. , 27(1):240–256, 2013.[6] P. Erd˝os and A. Gy´arf´as. Split and balanced colorings of complete graphs. DiscreteMath. , 200(1-3):79–86, 1999. Paul Erd˝os memorial collection.[7] W. Goddard and M. Henning. Thoroughly distributed colorings. arXiv:1609.09684 .[8] J. Goldwasser and R. Hansen. Polychromatic colorings of 1-regular and 2-regular sub-graphs of complete graphs. In preparation, 2017.[9] J. Goldwasser, B. Lidick´y, R. Martin, D. Offner, J. Talbot, and M. Young. Polychro-matic colorings on the hypercube. arXiv:1603.05865, 2016.[10] A. Gy´arf´as. Problems and memories. arXiv: 1307.1768v1 . Paul Erd˝os memorial collec-tion.[11] D. Offner. Polychromatic colorings of subcubes of the hypercube. SIAM J. DiscreteMath. , 22(2):450–454, 2008.[12] J. Pach and G. T´oth. Decomposition of multiple coverings into many parts.