Graph polynomials and group coloring of graphs
Bartłomiej Bosek, Jarosław Grytczuk, Grzegorz Gutowski, Oriol Serra, Mariusz Zając
aa r X i v : . [ m a t h . C O ] D ec GRAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS
BARTŁOMIEJ BOSEK, JAROSŁAW GRYTCZUK, GRZEGORZ GUTOWSKI, ORIOL SERRA,AND MARIUSZ ZAJĄC
Abstract.
Let Γ be an Abelian group and let G be a simple graph. We say that G is Γ -colorable if for some fixed orientation of G and every edge labeling ℓ : E ( G ) → Γ , thereexists a vertex coloring c by the elements of Γ such that c ( y ) − c ( x ) = ℓ ( e ) , for every edge e = xy (oriented from x to y ).Langhede and Thomassen [15] proved recently that every planar graph on n vertices has atleast n/ different Z -colorings. By using a different approach based on graph polynomials,we extend this result to K -minor-free graphs in a more general setting of field coloring .More specifically, we prove that every such graph on n vertices is F - -choosable, whenever F is an arbitrary filed with at least elements. Moreover, the number of colorings (for everylist assignment) is at least n/ . Introduction
Let Γ be an Abelian group and let G be a simple graph. We say that G is Γ -colorable iffor some orientation of G and every edge labeling ℓ by the elements of Γ there exists a vertexcoloring c by the elements of Γ such that c ( y ) − c ( x ) = ℓ ( e ) , for every edge e = xy (orientedfrom x towards y ). This notion was introduced by Jaeger, Linial, Payane and Tarsi [11] as adual concept to group connectivity .Answering a question posed in [11], Lai and Zhang [13] proved that every planar graphis Z -colorable. Recently, Langhede and Thomassen [15] strengthened this result by provingthat the number of Z -colorings of every planar graph on n vertices (for a fixed edge labeling)is at least n/ . The proof is elementary but quite involved.In this paper we further extend these results by using the polynomial method. It will beconvenient to introduce a slightly more general setting. Let F be an arbitrary field and let G be a simple graph. Suppose that each edge e = xy of G is assigned with a triple ( a, b, c ) ∈ F ,with a, b = 0 . We say that G is F -colorable if for every such edge labeling there exists avertex coloring f by the elements of F such that af ( x ) + bf ( y ) + c = 0 , for every edge e = xy .Clearly, F -colorability of a graph implies its Γ -colorability, where Γ is the additive group ofthe field F .Define a graph G to be F - k -choosable if it is F -colorable from arbitrary lists of elementsof F , each of size k , assigned to the vertices. Supported by the Polish National Science Center, Grant Number: NCN 2019/35/B/ST6/02472.
Theorem 1.
Let G be a graph on n vertices without a minor of K . Let F be an arbitraryfield with at least elements. Then G is F - -choosable. Moreover, the number of colorings(for a fixed list assignment and a fixed edge labeling) is at least n/ . The proof is based on the method of graph polynomials. We use two tools, the Combina-torial Nullstellensatz of Alon [3] and a result of Alon and Füredi [4] concerning the numberof non-zeros of polynomials in multidimensional grids.Notice that Theorem 1 only covers Abelian groups which are additive groups of a field. Fora general Abelian group Γ of order at least , Chuang, Lai, Omidi, Wang, and Zakeri provedin [8] by elementary methods that every K -minor free graph is Γ -5-choosable. Theorem 1does not extend this result but in the overlapping cases guarantees a stronger conclusion.2. The results
Graph polynomials.
Let G be a simple graph on the set of vertices V = { x , x , . . . , x n } .Let P G be the graph polynomial of G , defined by(2.1) P G ( x , x , . . . , x n ) = Y x i x j ∈ E ( x i − x j ) . We identify symbols denoting vertices of G with variables of P G . We may consider P G as a polynomial over an arbitrary field F . Notice that P G is defined uniquely up to thesign, depending on the choice between the two possible expressions, ( x i − x j ) or ( x j − x i ) ,representing the edge x i x j in the product (2.1). We shall assume that this choice is fixed.In the process of expanding the polynomial P G , one creates monomials by picking onevariable from each factor ( x i − x j ) . Thus, every monomial corresponds to the unique orien-tation of G obtained by directing the edge x i x j towards the picked variable. In consequence,the degrees of variables in the monomial coincide with the in-degrees of the vertices in thecorresponding orientation.Let M G denote the multi-set of all monomials arising in this way. So, the cardinality of M G is equal to m , where m = | E ( G ) | , and the multiplicity of each monomial M is equal tothe number of orientations of G sharing the same in-degree sequence (corresponding to thedegrees of variables in M ). The sign of a monomial M ∈ M G is the product of signs of allvariables picked to form M . The coefficient of a monomial M in P G , denoted as c M ( P ) , isthe sum of signs of all copies of M in M G . A monomial M is called non-vanishing in P G if c M ( P ) = 0 .Suppose now that to each (oriented) edge e = x i x j of a graph G is assigned an arbitrarypair ( a, b ) of non-zero elements of F . We say that the edges of G are decorated with pairs ( a, b ) , and we define the corresponding decorated graph polynomial D G in which every factor ( x i − x j ) is substituted with ( ax i + bx j ) : RAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS 3 (2.2) D G ( x , x , . . . , x n ) = Y x i x j ∈ E ( G ) ( ax i + bx j ) . Of course, different decorations may give different polynomials, but we will denote thewhole family of them with the same symbol D G , hoping that this ambiguity will not causetoo much confusion.2.2. Combinatorial Nullstellensatz.
For a monomial M , let deg x i ( M ) denote the degreeof the variable x i in M . The total degree of the monomial M is the sum P ni =1 deg x i ( M ) . Ina graph polynomial all monomials have the same total degree, which is equal to the numberof edges of G . Recall that the degree of a polynomial is the maximum of total degrees of itsnon-vanishing monomials.We will use the following famous theorem of Alon [3]. Theorem 2 (Combinatorial Nullstellensatz, [3]) . Let P be a polynomial in F [ x , x , . . . , x n ] ,where F is an arbitrary field of coefficients. Suppose that there is a non-vanishing monomial x k x k · · · x k n n in P whose total degree is equal to the degree of P . Then, for arbitrary sets A i ⊆ F , with | A i | = k i + 1 , there exist elements a i ∈ A i such that P ( a , a , . . . , a n ) = 0 . In view of this theorem it will be convenient to denote by A F ( P ) the least integer k suchthat the polynomial P has a non-vanishing monomial M whose degree is equal to the degreeof P and satisfying deg x i ( M ) k , for each i = 1 , , . . . , n .For the decorated graph polynomial D G , which is actually a collection of polynomials, wedefine A F ( D G ) as the least number k such that A F ( P ) k for every P in D G . It is not hardto demonstrate that for every graph G we have A F ( P G ) A F ( D G ) col( G ) − , where col( G ) is the coloring number of G , defined as the least k such that the vertices of G can be linearly ordered so that each vertex v has at most k − neighbors that precede v inthe order.2.3. Field coloring and graph polynomials.
Let F be any field and let ℓ be any edgelabeling of a graph G by the elements of F . Let D G be a decorated graph polynomial withsome fixed decoration over F . Consider the polynomial D G,ℓ defined as:(2.3) D G,ℓ ( x , x , . . . , x n ) = Y e = x i x j ∈ E ( G ) ( ax i + bx j + ℓ ( e )) . Our basic observation is formulated as follows.
BARTŁOMIEJ BOSEK, JAROSŁAW GRYTCZUK, GRZEGORZ GUTOWSKI, ORIOL SERRA, AND MARIUSZ ZAJĄC
Theorem 3.
Let G be any simple graph, with a graph polynomial P G , and let F be an arbitraryfield. Let ℓ be any edge labeling of G by the elements of F . Then A F ( D G,ℓ ) = A F ( D G ) .Proof. First notice that the polynomial D G,ℓ can be written as:(2.4) D G,ℓ = D G + Q, where Q is a polynomial of degree strictly smaller than the degree of D G . Indeed, we obtainthe summand D G by choosing the whole expression ( ax i + bx j ) in every factor of D G,ℓ . Inevery other case, the formed monomial will use at least one constant ℓ ( e ) , hence the totaldegree of such monomial will be strictly smaller than the number of edges in G , which isequal to the degree of D G . This means that every non-vanishing monomial of D G cannotvanish in D G,ℓ . This completes the proof. (cid:3)
Field coloring of planar graphs.
In [26] Zhu proved that every planar graph G satisfies A Q ( P G ) , though the same proof works for arbitrary field. This constitutes analgebraic analog of the famous result of Thomassen [19] on -choosability of planar graphs.We will derive below a slightly stronger statement. Theorem 4.
Let G be a planar graph and let D G be its decorated graph polynomial over anarbitrary field F . Then A F ( D G ) . By Theorems 3 and 4 we get immediately the following result.
Corollary 1.
Every planar graph is F - -choosable, where F is an arbitrary field with elements. The original proof from [26] is by the induction with the same scenario as in the famousThomassen’s proof from [19], except for one unexpected twist. We will give below a purelyalgebraic proof along similar lines of the following more general statement, stressing the factthat it works over an arbitrary field (which is crucial for our applications).
Theorem 5.
Let G be a near-traingulation and let e = xy be an arbitrary edge of theboundary cycle of G . Then a decorated graph polynomial D G − e over an arbitrary field F contains a non-vanishing monomial M satisfying the following conditions: (i) deg x ( M ) = deg y ( M ) = 0 , (ii) deg v ( M ) , for every boundary vertex v , (iii) deg u ( M ) , for every interior vertex u .Proof. Let us call a non-vanishing monomial M satisfying conditions (i)-(iii), a nice monomialfor ( G, e ) . We will use induction on the number of vertices of G . It is easy to check that G = K satisfies the assertion. In this case we have D G − e = ( a x + b v )( a y + b v ) , and theonly nice monomial is M = v whose coefficient is b b = 0 . Hence, M is non-vanishing.We distinguish two cases. RAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS 5
Case 1. G has a chord. Suppose first that G has a chord f = wz . This chord splits G into two subgraphs G and G . We assume that e is in G , while f belongs to both subgraphs. By induction we assumethat both graphs contain nice non-vanishing monomials M and M for ( G , e ) and ( G , f ) ,respectively.Let us denote P = D G − e , P = D G − e , and P = D G − f . Then we have P = P P , and wesee that the monomial M = M M appears in the expansion of P . We claim that M is a nicemonomial for ( G, e ) . It is easy to see that M satisfies conditions (i)-(iii). To see that it isnon-vanishing, notice that M = M M is actually the only way of expressing the monomial M as a product of two monomials from P and P , respectively. This is because the onlycommon variables of P and P are w and z , and they do not occur in M . This shows that c M ( P ) = c M ( P ) · c M ( P ) = 0 , confirming that M is non-vanishing in the polynomial P . Case 2. G has no chord. Suppose that there is no chord in G . Let v = x be the neighbor of y on the boundary of G . Let t be the other neighbor of v on the outer face, and let x , x , . . . , x k be the neighborsof v lying in the interior of G . Let G ′ = G − v .By the inductive assumption, there is a nice monomial M ′ for ( G ′ , e ) in the graph polyno-mial P ′ = D G ′ − e . So, we have c M ′ ( P ′ ) = 0 . Subcase 1.
Boundary face is a triangle.
Suppose first that t = x , which means that the boundary face is a triangle. In this casethe nice monomial M ′ has the form M ′ = Y x r x r . . . x r k k , where r i for each i = 1 , , . . . , k , and Y is a monomial consisting of the rest of variables.Since M ′ is nice for ( G ′ , e ) , the monomial Y does not contain variables x and y , and eachother variable z in Y satisfies deg z ( Y ) .First notice that P = P ′ Q , where Q = ( av + bx )( cv + dy )( a v + b x )( a v + b x ) . . . ( a k v + b k x k ) . In the expansion of Q we get monomial N = v x . . . x k , whose coefficient is c N ( Q ) = acb · · · b k = 0 . Hence, in the expansion of the product P ′ Q wewill get monomial M = M ′ N , which can be written as M = Y x r +11 x r +12 . . . x r k +1 k v . BARTŁOMIEJ BOSEK, JAROSŁAW GRYTCZUK, GRZEGORZ GUTOWSKI, ORIOL SERRA, AND MARIUSZ ZAJĄC
Clearly M satisfies conditions (i)-(iii). We claim that it is also non-vanishing in P . We claimeven more, namely, that there is only one way of expressing M as a product M = AB of twomonomials, with A from P ′ and B form Q . Indeed, to get v in B we have to use the firsttwo factors of Q , since otherwise we will have x or y in M . This implies that B = N and inconsequence A = M ′ . Thus c M ( P ) = c M ′ ( P ′ ) · c N ( Q ) = c M ′ ( P ′ ) = 0 , which shows that M is non-vanishing in P . Subcase 2.
There is a special monomial.
Assume now that t = x . Suppose first that there exists a non-vanishing special monomial S in P ′ which satisfies all conditions (i)-(iii), except that deg t ( S ) and deg x i ( S ) forat most one i . We may assume without loss of generality that i = 1 . So, we assume that c S ( P ′ ) = 0 .This special monomial S can be written in the form S = Zx s x s . . . x s k k t s , where s , s , s i for i = 2 , , . . . k , and Z is some monomial on the rest of variables.As before we may write the polynomial P = D G − e as the product P = P ′ Q with Q = ( av + by )( cv + dt )( a v + b x ) . . . ( a k v + b k x k ) . In the expansion of Q we get monomial N ′ = vtx . . . x k , with coefficient c N ′ ( Q ) = adb · · · b k = 0 . Hence, in the expansion of the product P ′ Q we willget the monomial M = SN ′ , which can be written as M = Zx s +11 x s +12 . . . x s k +1 k t s +1 v. Clearly, M satisfies conditions (i)-(iii). Also, as in the previous case, the splitting M = SN ′ is unique. Indeed, assume that M = AB is any decomposition of M into the product ofmonomials from P ′ and Q , respectively. To get v in B we have to use the first factor of Q ,since otherwise we will have y in M . This already implies that B = N ′ and in consequence A = S . Hence, c M ( P ) = c S ( P ′ ) · c N ′ ( Q ) = c S ( P ′ ) = 0 , so, M is non-vanishing in P . Subcase 3.
There is no special monomial.
RAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS 7
Finally, assume that there is no special monomial in P ′ . However, by inductive assumption,there is still a nice non-vanishing monomial M ′ in the graph polynomial P ′ = D G ′ − e . Thismonomial can be written now as M ′ = Y x r x r . . . x r k k t r , where r , r i , for all i = 1 , , . . . , k , and Y is a monomial consisting of the rest ofvariables.As in Subcase 1, we have P = P ′ Q , where Q = ( av + by )( cv + dt )( a v + b x )( a v + b x ) . . . ( a k v + b k x k ) . In the expansion of Q we get the monomial N = v x . . . x k , with c N ( Q ) = acb · · · b k = 0 . Hence, in the expansion of the product P ′ Q we will getmonomial M = M ′ N , which can be written as M = Y x r +11 x r +12 . . . x r k +1 k t r v . Clearly M satisfies conditions (i)-(iii). We claim that there is only one way of expressing M as a product of two monomials, M = AB , with A from P ′ and B form Q . Indeed, to get v in B we have to choose variable v exactly twice from factors of Q . The first choice mustbe from the first factor, otherwise y will appear in M . The second choice must be from thesecond factor, since otherwise the variable t will appear in B , while some x i will be missing.Then, in order to get M = AB , we would have to have t r − and x r i +1 i in the monomial A .But then A will be a special monomial in P ′ , contrary to our assumption. Hence, we musthave B = N and in consequence A = M ′ . Thus c M ( P ) = c M ′ ( P ′ ) · c N ( Q ) = c M ′ ( P ′ ) = 0 , which demonstrates that M is non-vanishing in P .The proof is complete. (cid:3) In [9] Grytczuk and Zhu proved that every planar graph G contains a matching S suchthat A F ( P G − S ) . The proof is similar to the above and can be easily modified to give thefollowing result. Theorem 6.
Every planar graph G contains a matching S such that A F ( D G − S ) , for anarbitrary field F . In consequence, G − S is F - -choosable, and in particular, Z × Z -colorable. K -minor-free graphs. To extend the above results to graphs without a K -minor wewill use the well-known characterization theorem of Wagner [23]. A similar approach wasmade by Abe, Kim, and Ozeki [1] in an extension of the result of Zhu [26] to graphs with no K -minor. BARTŁOMIEJ BOSEK, JAROSŁAW GRYTCZUK, GRZEGORZ GUTOWSKI, ORIOL SERRA, AND MARIUSZ ZAJĄC
Recall that a k -clique-sum of two graphs is a new graph obtained by gluing the two graphsalong a clique of size k in each of them, and possibly deleting some edges of the clique. Recallalso that the Wagner graph V is the graph obtained from the cycle C by adding four edgesjoining antipodal pairs of vertices. Theorem 7 (Wagner, [23]) . Every edge-maximal graph without a K -minor can be builtrecursively from planar triangulations and the graph V by clique-sums with cliques on atmost vertices. We will need the following result.
Theorem 8.
Let G be a plane triangulation and let T be any triangle in G . Then the deco-rated graph polynomial D G − E ( T ) over an arbitrary field F contains a non-vanishing monomial N such that deg u ( N ) = 0 , for every vertex u ∈ V ( T ) , and deg w ( N ) , for all other vertices.Proof. Let V ( T ) = { x, y, v } and denote e = xy . Suppose first that T is a facial triangle.We may assume that T is the outer face of G . By Theorem 5 we know that there is anon-vanishing monomial M in D G − e such that deg x ( M ) = deg y ( M ) = 0 , deg v ( M ) , and deg w ( M ) , for all other variables. Actually, to form this monomial we have to pick v exactly twice; once from every factor, ( ax + bv ) and ( cy + dv ) , since we can choose neither x ,nor y . Thus, when we delete the corresponding two edges xv and yv from the graph G − e ,we must have a non-vanishing monomial N = M/v in the polynomial D G − E ( T ) .If T is not a facial triangle, then we may split the triangulation G into two sub-triangulations, G and G , lying inside and outside the triangle T , respectively. Then we may apply thesame argument to each sub-triangulation separately to get the desired monomials N and N in polynomials D G − E ( T ) and D G − E ( T ) , respectively. Clearly, we have D G − E ( T ) = D G − E ( T ) D G − E ( T ) , and it is easy to see that N = N N is a non-vanishing monomial in D G − E ( T ) satisfying theassertion of the theorem. (cid:3) Now we may give the proof of the aforementioned extension of Theorem 4.
Theorem 9.
Let G be a graph without a K -minor and let D G be its decorated graph poly-nomial over an arbitrary field F . Then A F ( D G ) .Proof. Let G be an edge-maximal K -minor-free graph. We will proceed by the induction onthe number of terms in a clique-sum giving G . So, suppose that G is k -clique-sum, k , oftwo graphs H and F , where H is a clique-sum with a smaller number of terms, while F isa plane triangulation or F = V . Assume by the induction that A F ( D H ) , and let M bethe monomial witnessing this inequality with coefficient c M ( D H ) = 0 .In the triangulation case, let { x, y, z } be the three vertices of the common triangle T in H and F . Let N be a monomial in D F − E ( T ) guaranteed by Theorem 8 with coefficient RAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS 9 c N ( D F − E ( T ) ) = 0 . We claim that the monomial M N occurs in the polynomial D G withcoefficient c MN ( D G ) = c M ( D H ) · c N ( D F − E ( T ) ) . Indeed, we have an obvious equality D G = D H D F − E ( T ) and the only common variables ofthe two polynomial factors are x, y, z , none of which appears in the monomial N .If F = V , then the reasoning is similar. Notice that V is triangle-free, so the clique-sumcan be made on one vertex or one edge. Suppose it is the latter situation (the former is eveneasier). Let x, y be the two common vertices of H and F . It is enough to notice that for everyedge e = xy of V there is an acyclic orientation of V − e with in-degrees of both vertices x and y equal to . The monomial J corresponding to this orientation has coefficient equalto , the variables x and y do not occur in J , while other variables have degrees at most .Thus, as before we have c MJ ( D G ) = c M ( D H ) · c J ( D F − e ) = c M ( D H ) . This completes the proof. (cid:3)
By Theorems 9 and 3 we get immediately the following results, whose special case extends Z -colorability of planar graphs. Corollary 2.
Let F be an arbitrary field with at least elements. Then every graph G withouta K -minor is F - -choosable. The number of colorings.
In this section we prove the second part of Theorem 1.Our main tool is the following general result of Alon and Füredi [4].
Theorem 10 (Alon and Füredi, [4]) . Let F be an arbitrary field, let A , A , . . . , A n be anynon-empty subsets of F , and let B = A × A × · · · × A n . Suppose that P ( x , x , . . . , x n ) isa polynomial over F which does not vanish on all of B . Then the number of points in B forwhich P has a non-zero value is at least min Q ni =1 q i , where the minimum is taken over allintegers q i such that q i | A i | and P ni =1 q i > P ni =1 | A i | − deg P . For a convenient use of this result, and for the sake of completeness, we will prove aslightly weaker statement by an argument resembling a beautiful proof of CombinatorialNullstellensatz, due to Michałek [16]. A similar approach was made by Bishnoi, Clark,Potukuchi, and Schmitt [6] to get some generalization of the Alon-Füredi theorem.We need a simple technical lemma.
Lemma 1.
Let a , a , . . . , a n be positive integers, with max a i = t > and P ni =1 a i = S .Then (2.5) A = n Y i =1 a i > t S − nt − . Proof.
The proof is by the induction on n . For n = 1 we have A = a = t and S = a = t ,hence we get equality in (2.5). For n > , let a i = min a i = m . Then, by the inductiveassumption, we have Aa i = Am > t ( S − m ) − ( n − t − . Observe that for every x ∈ [1 , t ] we have(2.6) x > t x − t − . Indeed, it is not hard to check that the function f ( x ) = t ( x − / ( t − is convex, with f (1) = 1 and f ( t ) = t . Hence, taking x = m , we may write A > m · t ( S − m ) − ( n − t − > t m − t − · t ( S − m ) − ( n − t − = t S − nt − , as asserted. (cid:3) Theorem 11.
Let F be an arbitrary field, and let A , A , . . . , A n be any non-empty subsetsof F , with S = P ni =1 | A i | and t = max | A i | . Let B = A × A × · · · × A n and suppose that P ( x , x , . . . , x n ) is a polynomial over F of degree deg P = d , which does not vanish on all of B . Then the number of points in B for which P has a non-zero value is at least t S − n − dt − , provided that s > n + d and t > .Proof. The proof is by the induction on d and n . If d = 0 , then P equals some non-zeroconstant c ∈ F , and therefore all points of B are non-vanishing for P . There are exactly Q ni =1 | A i | of them, so, the assertion follows from Lemma 1 by putting a i = | A i | .For n = 1 and arbitrary d > , we know that the number of roots of a polynomial P is atmost d . Hence, the number of elements of A at which P is non-zero is at least t − d . So, bythe assumption that t > d + 1 and the inequality (2.6), we have t − d > t t − − dt − = t S − − dt − , as S = t in this case.Assume now that d > and n > . Let | A | = j , and assume, without loss of generality,that there is another set A i , with | A i | = t . Let b ∈ A be any element, and let us divide thepolynomial P by ( x − b ) : P = ( x − b ) Q b + R b . Observe that deg Q b = deg P − d − , deg R b d , and that the polynomial R b does notcontain the variable x .Suppose first that the polynomial R b vanishes at all points of the grid A × · · · × A n . Thisimplies that j > , since otherwise, the polynomial P would vanish over the whole grid B ,contrary to the assumption. Furthermore, each non-vanishing point of P in B is at the same RAPH POLYNOMIALS AND GROUP COLORING OF GRAPHS 11 time a non-vanishing point of Q b in the grid ( A − b ) × A × · · · × A n , and vice versa. Thus,by the inductive assumption we get that P has at least t ( S − − n − ( d − t − = t S − n − dt − non-vanishing points in B .Finally, suppose that for every b ∈ A , the polynomial R b has some non-vanishing pointsin the grid A × · · · × A n . Then each such point can be extended to a non-vanishing point of P by setting x = b . By the inductive assumption on n , the number of such points for each R b is at least t ( S − j ) − ( n − − dt − . Hence, the total number of non-vanishing points of P in the grid B is at least j · t ( S − j ) − ( n − − dt − > t j − t − · t ( S − j ) − ( n − − dt − = t S − n − dt − , by the inequality (2.6). The proof is complete. (cid:3) The above result and Theorem 9 easily imply the following corollary.
Corollary 3.
Let G be a graph on n vertices with no K -minor. Let F be an arbitrary fieldwith at least elements. Then for every assignment of lists of size from F to the verticesof G and every edge labeling ℓ , there are at least n/ different F -colorings of G from theselists.Proof. Let G be a graph satisfying assumptions of the corollary and let P G be its polynomial.Let A i ⊆ F be lists of size assigned to the vertices of G . So, keeping notation from Theorem11, we have t = 5 and S = 5 n . Since the number of edges in G is at most n − (by Theorem7), we have d = deg P G n − . Hence, the number of F -colorings is at least t ( S − n − d ) / ( t − > (5 n − n − (3 n − / = 5 ( n +6) / > n/ , as asserted. (cid:3) Discussion
Let us conclude the paper with some further observations and possible extensions of ourresults.First, let us point on other improvements of existing results one may easily get by usinggraph polynomials. For instance, in [20] Thomassen proved that every planar graph on n vertices has at least n/ colorings from arbitrary lists of size . By a direct application ofthe Alon-Füredi theorem one may improve this bound to n/ , arguing in the same way asin the proof of Corollary 3. A similar improvement could be made for planar graphs of girthat least . Thomassen proved that every such graph G with n vertices has at least n/ colorings from lists of size , while the Alon-Füredi theorem gives n/ . Another striking consequence can be formulated in connection to the famous theorem ofBrooks [7] which asserts that every connected graph G of maximum degree ∆ is ∆ -colorable,except for the two cases, when G is a clique or an odd cycle (see [24] for a short proof of somemore general versions). By Theorem 11 we get that the number of colorings in this case isat least ( √ ∆) n (1 − − ) . Another advantage of using graph polynomials is that any upper bound on A F ( P G ) notonly gives an upper bound for the choosability of G , but also for its game variant, knownas paintability or online choosability . Roughly speaking, a graph G is k -paintable , if Alice, acolor-blind person, can color the vertices of G from any lists of size k in a sequence of rounds;in each round a new color is highlighted in the lists and Alice must decide which vertices willbe painted by this color. This idea was introduced independently by Schauz [17] and Zhu[25]. It is clear that every k -paintable graph is k -choosable, but not the other way around.In [18] Schauz proved that every graph G satisfying A F ( G ) k − is k -paintable (see [10]for a different, purely algebraic proof). Hence, all our results for field choosability transferto field paintability in a direct way.Finally, let us mention about the intriguing conjecture relating group coloring and listcoloring of graphs, stated by Král’, Pangrác, and Voss in [12]. They suspect that if a graph G is Γ -colorable for every Abelian group of order at least k , then G is k -choosable. We offerthe following, somewhat provocative, strengthening of this statement. Conjecture 1.
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Email address : [email protected] Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662 Warsaw, Poland
Email address : [email protected] Theoretical Computer Science Department, Faculty of Mathematics and Computer Sci-ence, Jagiellonian University, 30-348 Kraków, Poland
Email address : [email protected] Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Email address : [email protected] Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-662 Warsaw, Poland
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