On multichromatic numbers of widely colorable graphs
aa r X i v : . [ m a t h . C O ] F e b On multichromatic numbers of widely colorable graphs
Anna Gujgiczer a,c
G´abor Simonyi b,a,a
Department of Computer Science and Information Theory,Faculty of Electrical Engineering and InformaticsBudapest University of Technology and Economics b Alfr´ed R´enyi Institute of Mathematics, Budapest c MTA-BME Lend¨ulet Arithmetic Combinatorics Research Group [email protected] [email protected] bstract
A coloring is called s -wide if no walk of length 2 s − s -widely colorable with t colors if and only if it admits a homomorphism into auniversal graph W ( s, t ). Tardif observed that the value of the r th multichromatic number χ r ( W ( s, t )) of these graphs is at least t + 2( r −
1) and equality holds for r = s = 2. Heasked whether there is equality also for r = s = 3. We show that χ s ( W ( s, t )) = t +2( s − s thereby answering Tardif’s question. We observe that for large r (with respectto s and t fixed) we cannot have equality and that for s fixed and t going to infinitythe fractional chromatic number of W ( s, t ) also tends to infinity. The latter is a simpleconsequence of another result of Tardif on the fractional chromatic number of generalizedMycielski graphs. Keywords: multichromatic number, homomorphism, Kneser graphs, generalized Mycielskigraphs, wide coloring.
Introduction
A vertex-coloring of a graph is called s -wide if the two endvertices of every walk of length2 s − s -wide if and only if the graph does not contain any odd cycle shorter than 2 s + 1. Theinteresting phenomenon is that some graphs have s -wide colorings that are also optimalcolorings.A 1-wide coloring is just a proper coloring. 2-wide colorings were first investigated byGy´arf´as, Jensen, and Stiebitz [5] who, answering a question of Harvey and Murty, showedthe existence of a t -chromatic graph for every t ≥ t -coloring in which the neighborhood of every color class is an independent set.The analogous statement including more distant neighborhoods is also proven in [5].3-wide colorings (that are called simply wide colorings in [17]) turned out to be relevantconcerning the local chromatic number of several graph families whose chromatic numbercan be determined by the topological method of Lov´asz [10], cf. [17] for more detailsand also for the relevance of s -wide colorability in the context of the circular chromaticnumber.A graph homomorphism from a graph F to a graph G is an edge-preserving map ofthe vertex set of F to the vertex set of G . The existence of such a map is denoted by F → G . It is easy to see that G → K t is equivalent to the t -colorability of graph G , thatis to χ ( G ) ≤ t , where χ ( G ) is the chromatic number of G . We refer to the book [7] for ageneral treatment of the theory of graph homomorphisms.Several other types of graph colorings can also be expressed by the existence of a graphhomomorphism to some target graph and s -wide colorability is no exception. It is provenindependently in [1] and [17] (and already in [5] for the s = 2 case) that s -wide colorabilitywith t colors is equivalent to the existence of a homomorphism to the following graph wedenote by W ( s, t ) as in [17]. V ( W ( s, t )) = { ( x . . . x t ) : ∀ i x i ∈ { , , . . . , s } , ∃ ! i x i = 0 , ∃ j x j = 1 } ,E ( W ( s, t )) = {{ ( x . . . x t ) , ( y . . . y t ) } : ∀ i | x i − y i | = 1 or x i = y i = s } . Proposition 1. ([1, 5, 17])
A graph G admits an s -wide coloring using t colors if andonly if G → W ( s, t ) . A different incarnation of the graphs W ( s, t ) appears in the papers [6, 21, 25], where(following Wrochna’s notation in [25]) a graph operation Ω k is given for every odd integer k and when applied to the complete graph K t for k = 2 s − W ( s, t ). We will give and make use of this alternative definition in Section 2.It is easy to see that W ( s, t ) can be properly colored with t colors: set the color ofvertex ( x . . . x t ) to be the unique i for which x i = 0. It is proven in [1, 5, 17] (cf. also thechromatic properties of the more general Ω k construction in [6, 21, 25]) that this coloringis optimal, that is, χ ( W ( s, t )) = t. (1)1his represents the surprising fact, that there are t -chromatic graphs that can be opti-mally colored in such a way, that the complete d -neighborhood of any color class is anindependent set for every d < s . (By d -neighborhood of a color class we mean the set ofvertices at distance exactly d from the closest element of the color class. In fact, if G is s -widely colored then not only the d -neighborhoods of color classes form independent setsfor d < s but all those vertices that can be attained via walks of length d from the givencolor class.) The proof of t -chromaticity of W ( s, t ) goes via showing that some othergraphs that are known to be t -chromatic admit a homomorphism into W ( s, t ). Thesegraphs include generalized Mycielski graphs, Schrijver graphs, and Borsuk graphs of ap-propriate parameters (for the definition of generalized Mycielski graphs see Section 3; cf.[15, 3, 12] for the definition of Schrijver graphs and Borsuk graphs and [17] for furtherdetails), showing in particular that all these graphs admit s -wide colorings. A commonproperty of all these graphs is that their chromatic number can be determined by thealready mentioned topological method introduced by Lov´asz in his celebrated paper [10]proving Kneser’s conjecture.For n, k positive integers satisfying n ≥ k the Kneser graph KG( n, k ) is defined on (cid:0) [ n ] k (cid:1) , the set of all k -element subsets of the n -element set [ n ] = { , , . . . , n } as vertex set.Two vertices are adjacent if and only if the k -element subsets they represent are disjoint.It is not hard to show that χ (KG( n, k )) ≤ n − k + 2 (for all n, k satisfying n ≥ k )and Kneser [8] conjectured that this estimate is sharp. This was proved by Lov´asz [10]thereby establishing the following result. Theorem (Lov´asz-Kneser theorem). χ (KG( n, k )) = n − k + 2 . For more about the topological method we refer to the excellent book by Matouˇsek [12].The existence of a homomorphism to the Kneser graph KG( n, k ) can also be interpretedas a coloring property: G → KG( n, k ) holds if and only if we can color the vertices of G with n colors in such a way that every vertex receives k distinct colors and if two vertices u and v are adjacent then the set of colors received by u is disjoint from the set of colorsreceived by v . Such colorings were first considered by Geller and Stahl, see [4, 18]. Stahl[18] introduced the corresponding chromatic number χ k ( G ) as the minimum number ofcolors needed for such a coloring, called a k -fold coloring and χ k ( G ) the k -fold chromaticnumber in [14] (or k -tuple chromatic number in [7]). The fractional chromatic number χ f ( G ) can be defined as χ f ( G ) = inf k (cid:26) χ k ( G ) k (cid:27) = inf n nk : G → KG( n, k ) o . Note the immediate consequence of this definition that if G → H then χ f ( G ) ≤ χ f ( H ).Not surprisingly, determining multichromatic numbers (that is, k -fold chromatic numbersfor various k ’s) can be even harder in general than determining the chromatic number2hich is the special case for k = 1. An example of this phenomenon is that while thechromatic number of Kneser graphs is already known by the Lov´asz-Kneser theorem, itis only a still open conjecture due to Stahl what homomorphisms exist and what do notbetween Kneser graphs, see Section 6.2 of [7] for details, cf. also [24].The starting point of our investigations was a question by Tardif [23] who observed that(1) combined with the Lov´asz-Kneser theorem implies that χ r ( W ( s, t )) ≥ t + 2( r −
1) (2)and that equality holds for r = s = 2. (This is also true in the case of r = s = 1 when itsimply means χ ( K t ) = t .) Tardif asked if there is equality also for r = s = 3. In particular,he was interested in whether W (3 , KG(12 ,
3) and/or W (3 , KG(11 ,
3) is true.Our main result will imply that this is actually not the case and equality does holdfor r = s = 3. The motivation for Tardif’s question came from recent developmentsconcerning Hedetniemi’s conjecture in which wide colorings also turned out to be relevant.Hedetniemi’s conjecture asked whether the so-called categorical (or tensor) product G × H satisfies χ ( G × H ) = min { χ ( G ) , χ ( H ) } . The conjecture is equivalent to say that G × H → K c implies that G → K c or H → K c must hold. (Although the latter directly only means χ ( G × H ) ≥ min { χ ( G ) , χ ( H ) } , the reverse inequality is essentially trivial by G × H → G and G × H → H following easily from the definition of the categorical product.) Ifthis holds for K c , then K c is called multiplicative. Hedetniemi’s conjecture formulatedin 1966 thus stated that K c is multiplicative for every positive integer c . This is trivialfor c = 1, easy for c = 2 and is a far from trivial result by El-Zahar and Sauer [2] for c = 3 published in 1985. For no other c it was decided (whether K c is multiplicative ornot) until 2019, when a breakthrough by Yaroslav Shitov took place who proved in [16]that the conjecture is not true by constructing counterexamples for large enough c ’s. Thesmallest c for which Shitov’s construction disproved the conjecture was extremely large(about 3 according to an estimate in [26]). This value was dramatically improved withina relatively short time. Using Shitov’s ideas in a clever way first Zhu [27] reduced c to 125,then developing the method further Tardif [23] showed a counterexample for c = 13. Heremarked that his construction would also work for c = 12 and 11, respectively, providedthat W (3 , KG(12 ,
3) and W (3 , KG(11 , Theorem 2. χ s ( W ( s, t )) = t + 2( s − . Later Wrochna [26] managed to improve on Tardif’s result using the ideas in [23] in adifferent way and proving that K c is not multiplicative for any c ≥ c = 4the only open case. (For more details about Hedetniemi’s conjecture see e.g. Tardif’ssurvey [22] and the more recent papers cited above.)The paper is organized as follows. We present the proof of Theorem 2 in Section 2. InSection 3 we elaborate on the problem of what we can say about χ r ( W ( s, t )) for general r .3t will be an immediate consequence of Theorem 2 combined with Tardif’s observation (2)that χ r ( W ( s, t )) = t + 2( r −
1) whenever r ≤ s . We will also observe that we cannot haveequality in (2) for large enough r . We will also show that the fractional chromatic numberof W ( s, t ) goes to infinity when t grows and s remains fixed. The paper is concluded withsome observations about the position of the graphs W ( s, t ) in the homomorphism orderof graphs. First we give the alternative definition of the graphs W ( s, t ) using the graph operationΩ k , where k = 2 ℓ + 1 is odd, that was already mentioned in the Introduction. We givethe definition of only Ω ℓ +1 ( K t ) that we will use and refer to [25] for the constructionΩ ℓ +1 ( G ) for general graphs G . Definition 1.
The graph Ω ℓ +1 ( K t ) is defined as follows. V (Ω ℓ +1 ( K t )) = { ( A , A , . . . , A ℓ ) : ∀ i A i ⊆ [ t ] , | A | = 1 , A = ∅ , ∀ i ∈ { , . . . , ℓ − } A i ⊆ A i +2 , A ℓ − ∩ A ℓ = ∅} ,E (Ω ℓ +1 ( K t )) = {{ ( A , A , . . . , A ℓ ) , ( B , B , . . . , B ℓ ) : ∀ i ∈ { , , . . . , ℓ − } A i ⊆ B i +1 , B i ⊆ A i +1 and A ℓ ∩ B ℓ = ∅} . Note that the above conditions also imply that A i − ∩ A i = ∅ for all 1 ≤ i ≤ ℓ whenever( A , A , . . . , A ℓ ) ∈ V (Ω ℓ +1 ( K t )).It is straightforward and well-known (see e.g. [25, 26]) that we have W ( s, t ) ∼ = Ω s − ( K t ) . Indeed, one can easily check that the following function g : V ( W ( s, t )) → V (Ω s − ( K t ))provides an isomorphism between W ( s, t ) and Ω s − ( K t ). g : ( x . . . x t ) ( A , A , . . . , A s − ) , where ∀ i ∈ { , , . . . , s − } : A i = { j : x j ≤ i and x j ≡ i mod 2 } . Remark 1.
We gave both descriptions of the graphs W ( s, t ), because we believe that bothare useful. In particular, we will formulate the proof of Theorem 2 using the descriptionof Ω s − ( K t ) as we believe that it makes the presentation of the proof easier to follow.4evertheless, when we were thinking about the proof we felt we could understand thestructure of these graphs better by considering its vertices as the sequences given in itsdefinition as W ( s, t ). (It is also remarked in [26] that it is the W ( s, t ) type descriptionfrom which one easily sees that the number of vertices is t ( s t − − ( s − t − ).) ♦ Next we recall Tardif’s observation (2) that we state as a lemma for further reference andalso prove for the sake of completeness.
Lemma 3. (Tardif [23])
For all positive integers r and sχ r ( W ( s, t )) ≥ t + 2( r − . Proof.
We cannot have W ( s, t ) → KG( t + h, r ) for h < r −
1) as χ (KG( t + h, r )) = t + h − r + 2 by the Lov´asz-Kneser theorem and this value is less then t = χ ( W ( s, t ))whenever h < r − (cid:3) Proof of Theorem 2.
We need to show χ s ( W ( s, t )) = χ s (Ω s − ( K t )) = t + 2( s − . Lemma 3 already shows that the right hand side is a lower bound thus our task is to provethe reverse inequality which is equivalent to the existence of a graph homomorphism from W ( s, t ) ∼ = Ω s − ( K t ) to KG( t + 2( s − , s ) . Below we give such a homomorphism f : ( A , A , . . . , A s − )
7→ { z , . . . , z s − } , where { z , . . . , z s − } ∈ (cid:0) [ t +2( s − s (cid:1) = V (KG( t + 2( s − , s )). To emphasize the map-ping for U = ( A , A , . . . , A s − ) we will also use the notation z i = f i ( U ) when f (( A , A , . . . , A s − )) = { z , . . . , z s − } . (Note that we do not assume that the z i ’s aremonotonically increasing with respect to their indices, we only need that all of them aredistinct for a given f ( U ) = { z , . . . , z s − } ).First assume that s ≥ s = 1 case is a trivial special case of (1).)For every even i ∈ { , . . . , s − } we consider the three sets A i − , A i − , A i and for each suchtriple we define two elements of f ( U ), namely f i − ( U ) = z i − and f i ( U ) = z i as follows.According to the relative sizes of these three sets we will decide which of the elements t + i − , t + i, ( t + s −
1) + i − t + s + i −
2, and ( t + s −
1) + i = t + s + i − f ( U ). For every even i we will either put two of these elements into f ( U ) or if not then we will find enough elements from [ t ] to compensate this hiatus. Thiswill give us s − f ( U ). Finally we will define f ( U ) as the missing s th element of f ( U ). The rules are as follows.i) If | A i − | > | A i − | then let f i − ( U ) = t + i − f i ( U ) = t + i . If | A i − | > | A i | , thenlet f i − ( U ) = t + s + i − f i ( U ) = t + s + i −
1. (Note that by A i − ⊆ A i at mostone of the above two inequalities can hold so our definition is meaningful.)5i) If | A i − | < | A i − | < | A i | , then we must have | A i \ A i − | ≥
2. In that case choose 2distinct elements of A i \ A i − (these will be elements from [ t ]) to be f i − ( U ) and f i ( U ).iii) If | A i − | < | A i − | = | A i | , then | A i \ A i − | ≥ . Let f i − ( U ) be an arbitrary element of A i \ A i − and let f i ( U ) = (cid:26) t + s + i − A i − ∪ A i ) ∈ A i − t + s + i − A i − ∪ A i ) ∈ A i .Note that since A i − ∩ A i = ∅ , f i ( U ) will be well defined.iv) If | A i − | = | A i − | < | A i | , then let f i − ( U ) = (cid:26) t + i − A i − ∪ A i − ) ∈ A i − t + i if min( A i − ∪ A i − ) ∈ A i − .Since A i − ∩ A i − = ∅ , f i − ( U ) is well defined. Let f i ( U ) be an arbitrary element of A i \ A i − . Such a choice is possible as A i \ A i − = ∅ in this case.v) If | A i − | = | A i − | = | A i | (which means A i = A i − ) then let f i − ( U ) = (cid:26) t + i − A i − ∪ A i − ) ∈ A i − t + i if min( A i − ∪ A i − ) ∈ A i − .Let f i ( U ) = (cid:26) t + s + i − A i − ∪ A i ) ∈ A i − t + s + i − A i − ∪ A i ) ∈ A i .vi) Finally, let f ( U ) be equal to the unique h ∈ A . Note that by the above we have defined f j ( U ) for every 0 ≤ j ≤ s − j = j ′ then f j ( U ) = f j ′ ( U ) thus we have f ( U ) ∈ V (KG( t + 2( s − , s ) as needed. We have to provethat f is indeed a graph homomorphism from W ( s, t ) ∼ = Ω s − ( K t ) to KG( t + 2( s − , s ) . We do this first and consider the case of even s (that will be similar) afterwards.Consider U = ( A , A , . . . , A s − ) and U ′ = ( B , B , . . . , B s − ). We have to show that if f ( U ) ∩ f ( U ′ ) = ∅ , then { U, U ′ } / ∈ E (Ω s − ( K t )) . Assume that f ( U ) ∩ f ( U ′ ) = ∅ and we have h ∈ f ( U ) ∩ f ( U ′ ) for some h ∈ [ t ]. Then wehave h appearing in some A j and some B k , where both j and k are even. In particular, h ∈ A s − ∩ B s − , thus A s − ∩ B s − = ∅ , therefore U and U ′ cannot be adjacent.Now assume that f ( U ) ∩ f ( U ′ ) = ∅ but the intersection is disjoint from [ t ] thus we have t + d ∈ f ( U ) ∩ f ( U ′ ) for some 1 ≤ d ≤ s − d is odd and d ≤ s −
1, then d = i − ≤ i ≤ s −
1, thus t + d ∈ f ( U ) means t + d = t + i − f i − ( U ). If this happens then either | A i − | > | A i − | or | A i − | = | A i − | and min( A i − ∪ A i − ) ∈ A i − . Similarly, t + d = t + i − ∈ f ( U ′ ) implies that either | B i − | > | B i − | or | B i − | = | B i − | and min( B i − ∪ B i − ) ∈ B i − . Assume for contradiction6hat { U, U ′ } is an edge of our graph Ω s − ( K t ). Then we must have A i − ⊆ B i − and B i − ⊆ A i − implying | A i − | ≤ | B i − | ≤ | B i − | ≤ | A i − | ≤ | A i − | , therefore we must have equality everywhere. By A i − ⊆ B i − and B i − ⊆ A i − (thatfollows from { U, U ′ } ∈ E (Ω s − ( K t ))) this implies A i − = B i − and B i − = A i − andtherefore j := min( A i − ∪ A i − ) = min( B i − ∪ B i − ) . Our assumption on d then impliesboth j ∈ A i − and j ∈ B i − = A i − which is impossible by A i − ∩ A i − = ∅ . The situation is similar for the other possible values of d . If d = i ≤ s − t + d = t + i ∈ f ( U ) ∩ f ( U ′ ) for some adjacent vertices U, U ′ would again imply | A i − | = | B i − | = | B i − | = | A i − | and thus A i − = B i − , B i − = A i − as above. Our assumption on d now would imply for j = min( A i − ∪ A i − ) = min( B i − ∪ B i − ) that it must be both in A i − and in B i − = A i − leading to the same contradiction as in the previous paragraph.For s − < d and t + d ∈ f ( U ) ∩ f ( U ′ ) for adjacent vertices U, U ′ we get the samecontradiction with the indices shifted by one. In particular, this assumption implies | A i − | ≥ | A i | and | B i − | ≥ | B i | that by the adjacency of U and U ′ (meaning, in particular, A i − ⊆ B i and B i − ⊆ A i ) would imply | A i − | = | B i | = | B i − | = | A i | and thus A i − = B i and B i − = A i . Then we obtain that k := min( A i − ∪ A i ) =min( B i ∪ B i − ) should belong (depending on the parity of d ) to both A i − and B i − = A i or to both A i and B i = A i − leading to the same contradiction that A i − ∩ A i = ∅ . Thisfinishes the proof for odd s .Now assume that s is even. We need only some minor modifications compared to the odd s case. Let us now for every odd i ∈ { , . . . , s − } define f i − ( U ) and f i ( U ) almost thesame way as in points i)– v) above. (The only difference will be that the values t + i − t + i are shifted by 1 to become t + i and t + i + 1. In case of s = 2 the modified rules(i’)-(v’) will not apply, only those will that we denote by (vi’) and (vii’) below.) Thisgives the last s − f ( U ) = { f ( U ) , f ( U ) , . . . , f s − ( U ) } , what is left is todefine f ( U ) and f ( U ) by a modified version of the sixth point above that has now twoparts. The modified rules are as follows.i’) If | A i − | > | A i − | then let f i − ( U ) = t + i and f i ( U ) = t + i + 1. If | A i − | > | A i | , thenlet f i − ( U ) = t + s + i − f i ( U ) = t + s + i − | A i − | = | A i − | < | A i | , then let f i − ( U ) = (cid:26) t + i if min( A i − ∪ A i − ) ∈ A i − t + i + 1 if min( A i − ∪ A i − ) ∈ A i − .7et f i ( U ) be an arbitrary element of A i \ A i − .v)’ If | A i − | = | A i − | = | A i | then let f i − ( U ) = (cid:26) t + i if min( A i − ∪ A i − ) ∈ A i − t + i + 1 if min( A i − ∪ A i − ) ∈ A i − .Let f i ( U ) = (cid:26) t + s + i − A i − ∪ A i ) ∈ A i − t + s + i − A i − ∪ A i ) ∈ A i .vi’) If | A | = | A | , then let f ( U ) = (cid:26) t + 1 if min( A ∪ A ) ∈ A t + 2 if min( A ∪ A ) ∈ A .Note that in this case both A and A contains only one element and the value of f ( U )is t + 1 or t + 2 depending on which of the two is smaller. At the same time let f ( U ) = h where A = { h } , that is, h ∈ [ t ] is the unique element of A .vii’) If | A | > | A | , then since | A | = 1 we have | A | ≥
2. Now choose two arbitrarydistinct elements of A for f ( U ) and f ( U ).Note that we have | A | ≥ | A | by the definition of Ω s − ( K t ), so we do not have toconsider the possibility that | A | > | A | , it never occurs.With this definition of f ( U ) the proof that f is a graph homomorphism is essentiallyidentical to that we presented in the odd s case. The main difference is that now those j ∈ [ t ] that appear as elements of the sets f ( U ) are all elements of some A i where i is odd,while the corresponding i ’s were all even in the case of odd s . The rest of the argumentswork the same way as in the case of odd s .This completes the proof. (cid:3) We remark that by the composition of homomorphisms Theorem 2 determines the s -foldchromatic number of every s -widely colorable t -chromatic graph. W ( s, t ) An immediate consequence of Theorem 2 is that we can give the multichromatic numbers χ r ( W ( s, t )) for all r ≤ s . Corollary 4. If r ≤ s , then χ r ( W ( s, t )) = t + 2( r − . Lemma 5. ([25])
For all ≤ r ≤ s we have W ( s, t ) → W ( r, t ) Proof.
Define the following function for all 0 ≤ a ≤ s . ϕ ( a ) = (cid:26) a if 0 ≤ a ≤ rr if r < a ≤ s .It is straightforward to check that the mapping g : ( x . . . x t ) ( ϕ ( x ) . . . ϕ ( x t )) is ahomomorphism from W ( s, t ) to W ( r, t ) for all 1 ≤ r ≤ s . (cid:3) Proof of Corollary 4
In view of Lemma 3 it is enough to prove that χ r ( W ( s, t )) is atmost the claimed value if r ≤ s . Applying Lemma 5 and Theorem 2 to r ≤ s we have W ( s, t ) → W ( r, t ) → KG( t + 2( r − , r )implying χ r ( W ( s, t )) ≤ t + 2( r − (cid:3) For r > s we do not know the value of χ r ( W ( s, t )). We know from Lemma 3 though that χ r ( W ( s, t )) ≥ t + 2( r −
1) so the question naturally arises whether we could have equalityhere for every r . Below we show that this is not the case. Proposition 6.
For all pairs of positive integers t ≥ and s ≥ there exists somethreshold r = r ( s, t ) > s for which χ r ( W ( s, t )) > t + 2( r −
1) (3) whenever r ≥ r .Proof. Assume for the sake of contradiction that for some fixed s and t we have χ r ( W ( s, t )) = t + 2( r −
1) for arbitrarily large r . That would imply that χ f ( W ( s, t )) ≤ lim r →∞ t +2( r − r = 2 . However, this cannot be true since W ( s, t ) is not bipartite for t ≥ C b +1 for some positive integer b . Thus we must have χ f ( W ( s, t )) ≥ χ f ( C b +1 ) = b +1 b , a number larger than 2 with the constant value b . (cid:3) The problem of determining the smallest possible r for which (3) holds is left as an openproblem. It is frustrating that we were not able to decide even whether this value is just s + 1 as the proof of Theorem 2 might suggest or larger. Remark 2.
The previous proof does not specify b as its value is not essential there.Nevertheless one can easily see that W ( s, ∼ = C s − . It is also easy to see that the odd9irth of W ( s, t ) must be at least 2 s +1 and we have equality here for t ≥ s +1 since a cycle C s +1 is formed in W ( s, s + 1) by the vertices given by the sequence (0 , , , . . . , s, s, s − , . . . , ,
1) and its cyclic permutations. (For larger t these sequences can be extended byan arbitrary number of coordinates equal to s .) In fact, the unpublished paper by Baumand Stiebitz [1] gives the general formula 2 s − (cid:6) s − t − (cid:7) for the odd girth of W ( s, t ). ♦ The previous proof raises the question what we can say about the fractional chromaticnumber of the graphs W ( s, t ). As a consequence of Theorem 2 we know χ f ( W ( s, t )) ≤ t +2( s − s and the previous simple proof implies that it is at least 2 + s − for t ≥ s the fractional chromatic number of W ( s, t ) getsarbitrarily large as t tends to infinity. Theorem 7.
For any fixed positive integer s we have lim t →∞ χ f ( W ( s, t )) = ∞ . The proof will be a simple consequence of the (already known) fact that certain generalizedMycielski graphs admit s -wide colorings. To give more details we introduce generalizedMycielski graphs below. Definition 2.
The h -level generalized Mycielskian M h ( G ) of a graph G is defined asfollows. V ( M h ( G )) = { ( v, j ) : v ∈ V ( G ) , ≤ j ≤ h − } ∪ { z } .E ( M h ( G )) = {{ ( u, i ) , ( v, j ) } : uv ∈ E ( G ) and ( | i − j | = 1 or i = j = 0 }∪{{ z, ( v, ( h − } . The d times iterated h -level generalized Mycielskian M h ( M h ( . . . M h ( G ) . . . )) of a graph G will be denoted by M ( d ) h ( G ) . The term Mycielskian of a graph G usually refers to M ( G ) = M ( G ) and Mycielski graphsare the iterated Mycielkians of K introduced by Mycielski [13] as triangle-free graphswhose chromatic number grows by one at every iteration. The property χ ( M ( G )) = χ ( G ) + 1 is well-known to hold for any G but the analogous equality is not always truefor h -level Mycielskians if h >
2, cf. Tardif [20]. Nevertheless Stiebitz [19] showed that χ ( M h ( G )) = χ ( G ) + 1 is also true if G is a complete graph or an odd cycle. (Moregenerally one can say that this is the case whenever G is a graph for which the topologicallower bound on the chromatic number by Lov´asz [10] is sharp, cf. [5, 12] or [17] for moredetails.) So by Stiebitz’s result we have χ ( M ( d ) h ( K )) = d + 2for all positive integers d and h . 10he t -chromaticity of W ( s, t ) is proven in [1, 5, 17] by showing the existence of t -chromaticgraphs that admit a homomorphism into W ( s, t ). In case of [1, 5] these are generalizedMycielski graphs M ( t − h ( K ) for appropriately large h . (Since [1] is unpublished and [5]gives this explicitly only for s = 2, we give some more details for the sake of complete-ness. Nevertheless, this is a straightforward generalization of the construction given in[5] as already noted in [17] where the case s = 3 is made explicit. So the following is astraightforward extension of Lemma 4.3 from [17] also attributed to [5] there.) Lemma 8. ([5]) If G has an s -wide coloring with t colors, then M s − ( G ) has an s -widecoloring with t + 1 colors.Proof. Fix an s -wide coloring c : V ( G ) → [ t ] of G . Let c : V ( M s − ( G )) → [ t ] ∪ { γ } bethe following coloring using the additional color γ . Set c ( z ) = γ and c (( v, j )) = (cid:26) γ if j ∈ { s, s + 2 , . . . , s − } c ( v ) otherwise.If we have a walk of odd length between vertices ( u, i ) and ( v, j ) with c ( u, i ) = c ( v, j ) ∈ [ t ]that walk must either traverse the vertex z or use an edge of the form { a, , ( b, } . Inthe latter case the walk projects down to a walk of the same length between u and v in G with c ( u ) = c ( v ) so its length must be at least 2 s + 1 by c being s -wide. In casethe walk traverses z we can assume that we have i j mod 2 and thus without loss ofgenerality j ≡ s mod 2 implying that j ≤ s −
2. But then the distance between ( v, j ) and z is already at least 2 s , so the length of our walk is at least 2 s + 1.Since deleting the set of vertices { ( v, } v ∈ V ( G ) from M s − ( G ) the remaining inducedsubgraph is bipartite and γ appears only on one side of this bipartite graph, any oddlength walk between two vertices colored γ must use an edge of the form { ( u, , ( v, } .But the distance of any γ -colored vertex from such vertices is at least s , so such a walkalso cannot be shorter than 2 s + 1. Thus c is indeed an s -wide coloring. (cid:3) For M ( G ) = M ( G ) Larsen, Propp and Ullman [9] made the very nice observation, that χ f ( M ( G )) can be given by a simple function of χ f ( G ), namely χ f ( M ( G )) = χ f ( G ) + 1 χ f ( G ) . This was later generalized by Tardif for generalized Mycielskians.
Theorem 9. (Tardif [20]) χ f ( M h ( G )) = χ f ( G ) + 1 P h − i =0 ( χ f ( G ) − i . Note that Tardif’s theorem implies that χ f ( M ( d ) h ( G )) tends to infinity as d goes to infinityfor any fixed finite h . 11 roof of Theorem 7 The proof is already immediate by the foregoing. Lemma 8 andTardif’s Theorem 9 together imply that χ f ( W ( s, t + 1)) ≥ χ f ( M s − ( W ( s, t ))) = χ f ( W ( s, t )) + 1 P s − i =0 ( χ f ( W ( s, t )) − i , and this implies the statement. (cid:3) In view of Lemma 8 it may be interesting to note that while a generalized Mycielskian of W ( s, t ) admits a homomorphism into W ( s, t + 1), the latter also admits a (very natural)homomorphism into another generalized Mycielksian of W ( s, t ). Proposition 10. W ( s, t + 1) → M s ( W ( s, t )) . Proof.
We explicitly give the homomorphism. Let g (( x . . . x t +1 )) = (( x . . . x t ) , s − x t +1 ) if x t +1 > x . . . x t ) ∈ V ( W ( s, t ))((01 . . . , s −
1) if { i : x i = 1 } = { t + 1 } z if x t +1 = 0.(In fact, in the second case ((01 . . . , s −
1) can be substituted by an arbitrarily chosen(( y . . . y t ) , s −
1) for which ( y . . . y t ) ∈ V ( W ( s, t )).)It is straightforward to check that the given function is indeed a graph homomorphism. (cid:3) Thus we obtained that in the homomorphism order of graphs (cf. [7]) in which F (cid:22) G if and only if F → G we have W ( s, t + 1) sandwiched between two different generalizedMycialskians of W ( s, t ), in particular, M s − ( W ( s, t )) (cid:22) W ( s, t + 1) (cid:22) M s ( W ( s, t )) . This excludes the possibility that our upper bound t +2( s − s on χ f ( W ( s, t )) provided byTheorem 2 would be tight at least for all sufficiently large t , because then the differ-ence χ f ( W ( s, t + 1)) − χ f ( W ( s, t )) would be equal to s for large t contradicting Tardif’sTheorem 9.With a little more considerations we can also show that W ( s, t + 1) is actually strictly sandwiched between the above two generalized Mycielskians of W ( s, t ) if s > t > Proposition 11. If s ≥ , t ≥ then M s − ( W ( s, t )) ≺ W ( s, t + 1) ≺ M s ( W ( s, t )) . (4) For s = 1 all three graphs are isomorphic to K t +1 . For s > , t = 2 we have M s − ( W ( s, ∼ = C s − ∼ = W ( s, ≺ M s ( W ( s, ∼ = C s +1 . roof. It is well-known and easy to prove that if G is a vertex-color-critical graph (thatis, one from which deleting any vertex its chromatic number decreases) and χ ( M h ( G )) = χ ( G ) + 1, then M h ( G ) is also vertex-color-critical (see this e.g. as Problem 9.18 in thebook [11] for h = 2). It is shown independently both in [1] and [17] that W ( s, t ) isedge-color-critical for every s ≥ , t ≥
2. Thus all three graphs appearing in (4) arevertex-color-critical. Since they all have the same chromatic number this implies that anyhomomorphism that exists between any two of them should be onto. This also means thatif any two of them would be homomorphically equivalent, then those two should have thesame number of vertices, in particular, any homomorphism between them is a one-to-onemapping between their vertex sets. This is clearly not the case for the homomorphismgiven in the proof of Proposition 10 since several distinct vertices (their exact number is s t − ( s − t ) are mapped to the vertex z unless s = 1.If a homomorphism between M s − ( W ( s, t )) and W ( s, t + 1) was one-to-one then bythe edge-color-criticality of W ( s, t + 1) it cannot happen that we map two non-adjacentvertices of M s − ( W ( s, t )) to two adjacent ones of W ( s, t +1), since then deleting the latteradjacency we would still have a homomorphism but into a graph of smaller chromaticnumber. Thus such a homomorphism would then be an isomorphism, that is the twographs would be isomorphic which is clearly not the case if s > t >
2. (A quickway to see this is the following. The maximum degree of W ( s, t + 1) is 2 t − attained byvertices ( x . . . x t +1 ) for which |{ i : x i = 1 }| is equal to 1 or 2. The maximum degree of M s − ( W ( s, t )) is | V ( W ( s, t )) | = t ( s t − − ( s − t − ) that cannot be a power of 2 for s > t = 2.) The remaining cases in the statement are straightforward to check. (cid:3) We thank Claude Tardif for sharing with us already an early version of his paper [23] whichcontained his interesting question that became the starting point of our work presentedhere.This research was partially supported by the National Research, Development and Inno-vation Office (NKFIH) grants K–120706 and BME NC TKP2020 and also by the BME-Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC). The work of GS wasalso partially supported by the grants K–132696 and SSN-135643 of NKFIH Hungary.
References [1] Stephan Baum, Michael Stiebitz, Coloring of graphs without short odd paths betweenvertices of the same color class, (2005), unpublished manuscript.[2] Mohamed El-Zahar, Norbert Sauer, The chromatic number of the product of two4-chromatic graphs is 4,
Combinatorica , 5 (1985), 121–126.133] P´al Erd˝os, Andr´as Hajnal, On chromatic graphs, (Hungarian)
Mat. Lapok , 18 (1967),1–4.[4] Dennis Geller, Saul Stahl, The chromatic number and other functions of the lexico-graphic product,
J. Combin. Theory, Ser. B , 19 (1975), 87–95.[5] Andr´as Gy´arf´as, Tommy Jensen, Michael Stiebitz, On graphs with strongly indepen-dent colour-classes,
J. Graph Theory , 46 (2004), 1–14.[6] Hossein Hajiabolhassan, On colorings of graph powers,
Discrete Math. , 309 (2009),4299–4305.[7] Pavol Hell, Jaroslav Neˇsetˇril,
Graphs and Homomorphisms , Oxford University Press,New york, 2004.[8] Martin Kneser, Aufgabe 300, Jahresber. Deutsch., Math. Verein. 58 (1955) 27.[9] Michael Larsen, James Propp, Daniel Ullman, The fractional chromatic number ofMycielski’s graphs,
J. Graph Theory , 19 (1995), no. 3, 411–416.[10] L´aszl´o Lov´asz, Kneser’s conjecture, chromatic number, and homotopy,
J. Combin.Theory, Ser. A , 25 (1978), no. 3, 319–324.[11] L´aszl´o Lov´asz,
Combinatorial Problems and Exercises, nd Edition , Akad´emiai Kiad´o,Budapest and Elsevier, 1993.[12] Jiˇr´ı Matouˇsek,
Using the Borsuk-Ulam Theorem, Lectures on Topological Methods inCombinatorics and Geometry , Springer-Verlag, Heidelberg, 2007.[13] Jan Mycielski, Sur le coloriage des graphs,
Colloq. Math. , 3 (1955), 161–162.[14] Edward R. Scheinerman, Daniel H. Ullman,
Fractional Graph Theory , Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley and Sons,Chichester, 1997.[15] Alexander Schrijver, Vertex-critical subgraphs of Kneser graphs,
Nieuw Arch. Wisk.(3) , 26 (1978), no. 3, 454–461.[16] Yaroslav Shitov, Counterexamples to Hedetniemi’s conjecture,
Ann. Math. , 190(2019), 663-667, arXiv:1905.02167 [math.CO].[17] G´abor Simonyi, G´abor Tardos, Local chromatic number, Ky Fan’s theorem, andcircular colorings,
Combinatorica , 26 (2006), 587–626, arXiv:math.CO/0407075.[18] Saul Stahl, n-Tuple colorings and associated graphs,
J. Combin. Theory, Ser. B , 20(1976), 185–203. 1419] Michael Stiebitz, Beitr¨age zur Theorie der f¨arbungscritischen Graphen, Habilitation,TH Ilmenau, 1985.[20] Claude Tardif, Fractional chromatic numbers of cones over graphs,
J. Graph Theory ,38 (2001), 87–94.[21] Claude Tardif, Multiplicative graphs and semi-lattice endomorphisms in the categoryof graphs,
J. Combin. Theory, Ser. B , 95 (2005), 338–345.[22] Claude Tardif, Hedetniemi’s conjecture, 40 years later,
Graph Theory Notes N. Y. ,54 (2008), 46–57.[23] Claude Tardif, The chromatic number of the product of 14-chromatic graphs can be13, manuscript (available on researchgate.net), 2020.[24] Claude Tardif, Xuding Zhu, A note on Hedetniemi’s conjecture, Stahl’s conjectureand the Poljak-R¨odl function,
Electronic. J. Combin. , 26 (2019), P4.34.[25] Marcin Wrochna, On inverse powers of graphs and topological implicationsof Hedetniemi’s conjecture,
J. Combin. Theory, Ser. B , 139 (2019), 267–295,arXiv:1712.03196 [math.CO].[26] Marcin Wrochna, Smaller counterexamples to Hedetniemi’s conjecture,arXiv:2012.13558 [math.CO].[27] Xuding Zhu, Relatively small counterexamples to Hedetniemi’s conjecture,