CC BY
7
УДК 519.179.1+519.174+519.176 DOI: 10.53815/20726759_2022__14__1__49
В. В. Cherkashin1'2, A. S. Gordeev3'4
1Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B, Saint Petersburg
199178 Russia
2 Moscow Institute of Physics and Technology, Laboratory of Combinatorial and geometric structures 3St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of
Sciences
4The Euler International Mathematical Institute, St. Petersburg, Russia
On list chromatic numbers of 2-colorable hypergraphs
We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the Schauz bound on fc-p^tite fc-uniform hypergraphs. It makes sense for sparse hypergraphs. In particular we show that a fc-uniform fc-regular hypergraph has the list chromatic number 2 for k > 4. Also, we obtain both lower and upper bounds on the list chromatic number of a complete s-uniform 2-colorable hypergraph in the vein of the Erdos-Rubin-Taylor theorem.
Key words: list colorings, hypergraph colorings, Erdos-Rubin-Taylor theorem. Д.Д. Черкашин1'2, А. С. Гордеев3'4
1
2
структур
3
4
О списочных хроматических числах двудольных
гиперграфов
Мы доказываем верхнюю оценку на списочное хроматическое число двудольного гиперграфа, которая обобщает оценку Шауза на fc-дольные ^-регулярные гиперграфы. Эта оценка содержательна для разреженных гиперграфов: в частности, мы показываем, что списочное хроматическое число любого ^-однородного ^-регулярного гиперграфа равно 2 при k ^ 4. Также мы получаем нижнюю и верхнюю оценки на списочное хроматическое число полного s-однородного двудольного гиперграфа в духе теоремы Эрдёша-Рубина-Тейлора.
Ключевые слова: списочные раскраски, раскраски гиперграфов, теорема Эрдёша-Рубина-Тейлора.
1. Introduction
A hypergraph Н is a pair of sets (V, Е), wherе V is finite and E С 2V\ V is called the set of vertices and E the set of edges. A hypergraph is said to be n-uniform if all of its edges have size n (further we call them n-graphs). Note that a graph is a 2-uniform hypergraph.
Given a hypergraph H = (V, E) and a vertex v £ V, define the degree of the vertex dn(v) as the number of edges of H that contain v.
A coloring of a hypergraph with r colors is a map f : V ^ {1,..., r}. A coloring f of a hypergraph is said to be proper if each edge e £ E contains two vertices v1,v2 £ e such that
© Черкашин Д. Д., Гордеев А. С., 2022
© Федеральное государственное автономное образовательное учреждение высшего образования
«Московский физико-технический институт (национальный исследовательский университет)», 2022
f (vi) = f The minimal number r for which there exists a proper r-coloring of H is called the chromatic number x(H) of the hypergraph H.
Consider a 2-colorable hvpergraph H. Its proper 2-coloring gives a partition of V into two sets A U B = V, A n B = 0 such that every edge of H intersects both A and B. Further we call such sets A and B parts of H. For a subset of a part T c A define its neighborhood
nh(T)= U e \ A
e£E, en T =0
For convenience define also Nh(v) = Nh({^}}-
Define a complete 2-colorable hvpergraph as s-graph with sizes of parts n, m and all possible edges of size s intersecting both parts.
An orientation of a hvpergraph H = (V, E) is a map p : E ^ V which satisfies p(e) e e for anv e e E. For orientation p define a degree function dv on the set of vertices as follows: dv(v) = \p-l(v)\.
To each vertex v we assign a list L(v) of colors which can be used for v. Given a map f : V ^ N we say that a hvpergraph H is ^^^^^^^^orable (/-choosable) if for any set of lists with lengths \L(v)\ = f (v) there exists a proper coloring of H. Hvpergraph is fc-choosable if it is /-choosable, where f (v) = k for each v. The to ^^^^matic number ch(^) of the hvpergraph # is the minimum number k such that H is fc-choosable.
List colorings of graphs and hvpergraphs were independently introduced by Vizing [12] and by Erdos, Rubin, and Taylor [5]. Clearly, ch(#) > x(H), since all lists can be taken equal to {1,...,ch(#)}. At the same time, the chromatic number and the list chromatic number are different, for example, for the complete bipartite graph K3,3, for which ch(^3,3) = 3 (equality is attained at the lists {1,2}, {1,3}, {2,3} assigned to the vertices of each part).
Sparse case. For a hvpergraph H = (V, E) denote
\E '\
L(H) = max
0=E'CE | Uee£Y e|'
One of the methods of estimating the list chromatic number of a graph is the Alon-Tarsi method [3]: if there exists an orientation ip of a graph G with certain structural properties, then G is (dp + 1)-choosable. In the case of bipartite (2-colorable) graphs any orientation has the desired properties, which gives two following theorems.
Теорема 1. Let p be an orientation of a bipartite graph G = (V,E). Then G is (dv + 1)-choosable.
Теорема 2 (Theorem 3.2 in [3]). For any bipartite graph G, ch(G) < \L(G)] + 1.
In 2010 Schauz [10] generalized these results to fc-partite fc-uniform hypergraphs, i.e. k-uniform hvpergraphs whose set of vertices can be partitioned into к sets V\,...,Vk such that each edge has exactly one vertex in each of Vi (bipartite graphs correspond tо the case к = 2). We show that in fact the same results hold in even more general case of 2-colorable hvpergraphs.
Dense case. Let N(n, r) denote the minimum number of vertices in an n-partite (n-colorable) graph with the list chromatic number larger than r. The following classic theorem express the asvmptotics of N(2, r) in terms of the minimal number of edges in an n-uniform hvpergraph without a proper 2-coloring. The latter quantity is denoted bv m(n).
Теорема 3 (Erdos-Rubin-Tavlor [5]). For any r
m(r) < N(2, r) < 2m(r).
The problem of finding m(n) is well-known, the best current bounds are
схГ^2п < m(n) < (1 + o(1))^П2n22n. V In n~ 4
For details see survey [9].
Kostochka extended Theorem 3 in two ways. Let Q(n, r) be the minimal number of edges in an n-uniform n-partite hypergraph with the Aromatic number greater than r\ m(n, r) be the minimal number of edges in an n-uniform hypergraph with the chromatic number greater than r, and finally p(n, r) be the minimal number of edges in an n-uniform hypergraph without an r-coloring such that every edge meets every color.
Теорема 4 (Kostochka [8]). For every n,r > 2 the following inequalities hold
m(r, n) < Q(n, r) < nm(r, n); p(r, n) < N(n,r) < np(r, n).
Upper and lower bounds on the quantities m(n, r) and p(n, r) heavily depend on the relations between n and r. The picture is collected in survey [9] (see also more recent paper [1]).
2. Sparse case
We give an alternative, more direct proof of the next theorem in the Appendix, since it is rather concise.
Теорема 5. Let <p be an orientation of a 2-colorable hypergraph H = (V, E). Then H is (dv + 1)-choosable.
Доказательство. Since H is 2-colorable, we can choose a vertex ue £ e in each edge e such that (p(e) and ue belong to different parts of H.
Consider a bipartite graph Б on the set of vertices V with the set of edges Ueee{(^(e),ue)}-Consider the following orientation of Б <p'((<p(e),ue)) = <p(e). Note that (v) = d^(v) for any v £ V. Then, by Theorem 1, Б is (dv + 1)-choosable. Since any proper сoloring of Б is a proper coloring of H, it follows that H is (dv + 1)-choosable. □
The following lemma first appeared in [10] (see Lemma 3.2). We give a proof of it here for the sake of completeness.
Лемма 1. For any hypergraph H = (V, E) there exists an orientation ip of H with dv < \L(H)].
Доказательство. Consider a bipartite graph G with parts V x {1,..., \L(H)]} and E, in which (v, i) and e are connected if and only if v and e are incident in H. The statement of lemma is true if and only if there is a matching in G which covers E.
Let 0 = T С E be ад arbitrary subset of edges of H. We estimate the size of the neighborhood of Tin G:
\Ng(T)| = | UeeT e| ■ \L(H)! > | UeeT e| ■ = \T|.
| UeeT e\
By Hall's marriage theorem it follows that there exists a matching in G which covers E. □
Theorem 5 and Lemma 1 combined give the following theorem.
Теорема 6. For any 2-colorable hypegraph H, ch(H) < \L(H)] + 1.
The exact value of L(H) can be difficult to calculate, so we provide a bound in simpler terms. For a hypergraph H denote the m^imum degree of a vertex in H bv A(H) and the minimum size of an edge in H by s(H).
Следствие 7. Let H = (V., E) be a 2-colorable hypergraph. Then
- A(H )-
ch(#) <
Доказательство. For any E' С E we have
s(H )
+ 1.
IE'1- s(H) < £ |e| < | UeeB/ e|- A(H),
eeE'
SO
L(H) <
A(H ) s(H) •
□
For an arbitrary (not necessarily 2-colorable) hypergraph a weaker bound with an additional factor of 2 holds. The next theorem was essentially proved by Gravin and Karpov [6], though they were not considering their results in the context of list colorings. In fact, the theorem in [6] has the second part in which «+1» is removed under some assumptions; it can be also generalized on the list chromatic numbers.
Теорема 8. Let H = (V, E) be a hypergraph. Then
' 2A(H )
Доказательство. Let
ch(#) <
k =
s(H )
2A(H )
+ 1.
Let G be the incidence graph of i.e. a bipartite graph with parts V, E and with v G V, e G E connected in G if and only if v and e are incident in We want to delete some edges from G to obtain a graph K with the following conditions on degrees:
dK (e) = 2 for an y e G E ;
max dK (v) < k.
v£V
If there exists such K, ^rnote Nk(e) = {ve, ue} and consider a graph B on a set of vertices V with the set of edges Ue eE {(ve, ue)}- One can color vert ices of B in any order, each time using any color not vet taken by adjacent vertices, to obtain a bound ch(5) < A(B) + 1 < fc + 1. Since any proper coloring of B is a proper coloring of it follows that ch(#) < k + 1.
If there is no K with desired properties, then we choose such K = (V,Ek) that dK(e) = 2 for any e e E and p(K) = ^veV max(0, dK(v) — fc) is as small as possible. Denote by S c V the set of vertices with dK > k. Define an augmenting path as such a sequence of vertices and edges v\,e\,..., vm, em, vm+\ that Vi e V, ei e (vi, e^) e Ek, (v+\,ei) e Ek, a are pairwise distinct, v\ e 5. Let U c V ^e the subset of vertices of V reachable by an augmenting path. Note that:
• For any v e U the inequality dK (v) > fc holds (otherwise we could flip the status of edges on the corresponding augmenting path and decrease p(K)).
• For any e e £ if Nk(e)nU = 0 then \NG(e) \U\ ^ L Indeed, suppose that v e Nk(e) nU, then any w e Nc(e) \ Nk(e) ^to lie in U, because there is an augmenting path ending in W] it follows that the only vertex of Nq (e) which ^rn possibly not lie in U is the unique vertex from Nk(e) \ {v}.
Let T c NK (U) be a set of e e NK (U) such th at lNG(e) \U | = 1 Since dK (v) ^ k for any v e U and dK (v) > k fa anv v e S, we get the inequality
INK(U) \T|>kM.
Now we obtain the lower bound on the sum of degrees do over vertices of U by summing up lNG(e) n U| over e e NK(U):
J2dG(v) > \T\(s(H) - 1) +
veu
k\U \-\T \
s(H) > A(H)\U\ + \T^s(H) - 1 - ^H^ > ^H)\U\.
It follows that there exists such v G U with do(v) > A(H), which is a contradiction.
□
3. Dense case
H
1
t< 4(1 + *1/lУ.
Then
ch(H) < I.
Доказательство. Consider the set of lists as a hvpergraph F. Then t = \E(F)|; we still refer to edges of F as lists to avoid the confusion with edges of H. We split the palette V(F) randomly into three parts: blue colors, red colors and neutral colors in the following way. Every vertex of V (F) is red or blue uniformly and independently with probability , so with probability p it is neutral; the value of p will be defined later. Then the expectation of monochromatic lists is equal to
1 — рЛ
A = 2
M
\E(F )\.
We call a list dangerous, if it has no blue or no red vertices. The expectation of dangerous lists equals to
'1 + pN
В = 2
m
\E (F )\.
In the case A < 1/2 and B < s/2 Markov inequality implies that with positive probability F has no monochromatic edges and the number of dangerous lists is smaller than s. Recall that H is 2-colorable; under the assumption one can color vertices of one part of H in blue or neutral
H H
H H
s1/l -1 " '
P
1+ S1/1 '
В
then — = A
(1+1У
On the other hand,
Summing up,
A =
(îb) = (1+sl/')'
2\E (F )\
2
1 + l/ 1 + l/
1 2 '
hence
В = a( =AS<^, V1 - v) 2 '
so the condition on t implies ch(H) < I.
□
Obviously, ch(Kts/2t/2) < ch(Kt2/2t/2) = (1 + o(1))log21 bv Erdos-Rubin-Tavlor theorem. Note that for s = ta, where a is a constant, the bound in Theorem 9 is constant times better.
Теорема 10. Suppose that
i = Q((logs + logO ■ I2(1 + s12^) .
Then
ch(Kt/2,t/2) > I.
Доказательство. Denote H = К**/2 v = ^ consider a set of random (independent
and uniform) lists with size I over the left part of H, and its copy over the right part. Let F be the resulting random hvpergraph of colors; we refer to its edges as lists to avoid the confusion H
Suppose the contrary: in particular it means that for every F hvpergraph H = К./21/2 has proper coloring in colors of F. Consider ад arbitrary such proper coloring кн- We call a color (a vertex of F) poor, if it appears in both parts of H; so a poor color appears at most s — 1 times. Define a coloring kf от vertices of F as follows: if a color appears only in the left part
H F
H F
vertex has no color. Note that kf contains no monochromatic lists; indeed if a list is blue then
H
contradicts with the existence of кн ■ We show that with positive probability such a coloring kf cannot exist.
Consider an arbitrary vertex coloring к of F in two colors in which some vertices remain colorless. We call a list dangerous if it has no blue or no red vertices. It turns out that with positive probability every к has a monochromatic list or greater than v0(s — 1) dangerous lists, where v0 is the number of colorless vertices. Hence kf (and also кн) cannot exist, which gives the desired contradiction. Below we provide an upper bound on the probability of every к to have no monochromatic list and no more than v0(s — 1) dangerous lists.
Let с = r£0 be the ratio of colorless vertices, denote by p1 = р1(к) the probability that a random list is monochromatic and by p2 = р2(п) the probability that it is dangerous. By definition
pi=«м. P2.
Then the probability that we have at most (s — 1)cv dangerous lists is smaller than the probability that there are at most sI2 dangerous lists. The latter is evaluated by
ST (t/2)p2(1 — P2)t22-% < £ (tp2-Y e-^/2-) < £ (tp2-Y e-^/2-*12
Consider the following substitution с = /¡+1 and put T = /i+1y • We got
s ■ 1 г ■
(1 + c\l ( s1/l Y 2s
P2 = 2Ьт) = 2(/^TlJ
2 ) \s1/l + 1/ ( s1/1 + 1)r
Then = s T and
One has
1 e-P2(t/2sl2) < s ¿2^ tPPl^ * e-P2(t/2-sl2).
e-P2 (t/2-sl2) < e-(P2t/2-sl2) = e-.sT+sl2 .
Also
si2
s I2 ^j8 = (sT + o(1))sl2 = e^0«)log(sT)sl2 Summing up, under the conditions of theorem
(1 + o(1))l2 s log(sT) - sT + sI2 < 0
and the event that the number of dangerous lists is smaller than sI2 has probability smaller than 1 " 2'
On the other hand
__t 1
(1 - Pi)t/2 < e~P11/2 < e (i+«1/I)1 = e~T < 1.
So for c = a random ¿-graph of lists with positive probability has both a monochromatic
list and at least si2 dangerous lists in every 2-coloring. From a combinatorial argument for
2-coloring, and for bigger c with positive probability has at least sI2 dangerous lists in every 2-coloring.
□
4. Applications and discussion
k k a k-uniform k-regular hvpergraph is 2-colorable for k > 4 [7,11]. Thus Theorem 6 gives
k k
k > 4
2) It turns out that there is a large gap between the bounds in dense and sparse cases; the same holds even for 2-graphs. As far as we are aware, the best known general bounds on the choice number of d-regular bipartite graph G are the following (see [4])
(2 - 0(D) l0gi < ch(G) <
Note that Erdos-Rubin-Tavlor theorem gives a tight bound for a complete bipartite graph
ch(Kl) = (1 + o(1)) log t.
3) Theorems 9 and 10 can be extended for r-colorable hvpergraphs (r > 2 is a constant) in a direct way.
Acknowledgments. The research of Danila Cherkashin is supported by «Native towns», a social investment program of PJSC «Gazprom Neft». The research of Alexev Gordeev was funded by RFBR, project number 19-31-90081.
Appendix. Alternative proof of Theorem 5
Denote parts of H as A, B c V\ every edge of H intersects both A and B. For each edge e of H choose a spanning tree Te on its set of vertices such that each edge of Te connects a vertex from part A with a vertex from part B. Consider the following polynomial on variables {xa}aeA, { Ub} beB-
fh(x,y) = U I Y^ (Xa -yb)\ .
eeE \(a,b)eTe J
Note that if FH(x, y) = 0 then the values of x, y form a proper coloring of H (the reverse is not necessarily true). By Combinatorial Nullstellensatz (see [2]), if the coefficient
П XdaAa)U УГ
aeA ьев
dv{b)
Fh =0,
then H is (dv + 1)-choosable. Consider now
Fhy) = Fh(x, -y) = П I (xa + Vb)
eeE \(a,b)eTe
Note that
П xt(a) П yd{b)
.aeA
ьев
FH = ±
П xt{a) П yd/b)
.aeA
beB
F
h ,
hence one can study coefficients of F* instead of Fh. The coefficient of FH in question is nonzero, because no summands will cancel out after opening the brackets, and orientation p corresponds to at least one summand with the desired coefficient.
Литература
1. Akhmejanova M., Balogh J. Chain method for panchromatic colorings of hvpergraphs // arXiv preprint arXiv:2008.03827v3. 2020.
2. Alon N. Combinatorial Nullstellensatz // Combinatorics, Probability and Computing. 1999. V. 8, N 1-2. P. 7-29.
3. Alon N., Tarsi M. Colorings and orientations of graphs // Combinatorica. 1992. V. 12, N 2. P. 125-134.
4. Alon N. Degrees and choice numbers // Random Structures k, Algorithms. 2000. V. 16, N 4. P. 364-368.
5. Erdos P., Rubin A.L., Taylor H. Choosabilitv in graphs // Congr. Numer. 1979. V. 26. P. 125-157.
6. Gravin N. V., Karpov D. V. On proper colorings of hvpergraphs // Zap. Nauchn. Sem. POMI. 2011. V. 391. P. 79-89.
7. Henning M.A., Yeo A. 2-colorings in ^-regular fc-uniform hvpergraphs // European Journal of Combinatorics. 2013. V. 34, N 7. P. 1192-1202.
8. Kostochka A. On a theorem of Erdos, Rubin, and Taylor on choosabilitv of complete bipartite graphs // The Electronic Journal of Combinatorics. 2002. V. 9, N 9. P. 1.
9. Raigorodskii A.M., Cherkashin D.D. Extremal problems in hvpergraph colourings // Russian Mathematical Surveys. 2020. V. 75, N 1. P. 89-146.
10. Schauz U. A paintabilitv version of the combinatorial Nullstellensatz, and list colorings of fc-partite fc-uniform hvpergraphs // The Electronic Journal of Combinatorics. 2010. P. R176.
11. Thomassen C. The even cycle problem for directed graphs // Journal of the American Mathematical Society. 1992. V. 5, N 2. P. 217-229.
12. Vizing V.G. Vertex colorings with given colors // Diskret. Analiz. 1976. V. 29. P. 3-10.
References
1. Akhmejanova M., Balogh J. Chain method for panchromatic colorings of hvpergraphs. arXiv preprint arXiv:2008.03827v3. 2020.
2. Alon N. Combinatorial Nullstellensatz. Combinatorics, Probability and Computing. 1999. V. 8, N 1-2. P. 7-29.
3. Alon N., Tarsi M. Colorings and orientations of graphs. Combinatorica. 1992. V. 12, N 2. P. 125-134.
4. Alon N. Degrees and choice numbers. Random Structures k, Algorithms. 2000. V. 16, N 4. P. 364-368.
5. Erdos P., Rubin A.L., Taylor H. Choosabilitv in graphs. Congr. Numer. 1979. V. 26. P. 125157.
6. Gravin N. V., Karpov D. V. On proper colorings of hvpergraphs. Zap. Nauchn. Sem. POMI. 2011. V. 391. P. 79-89.
k k
of Combinatorics. 2013. V. 34, N 7. P. 1192-1202.
8. Kostochka A. On a theorem of Erdos, Rubin, and Taylor on choosabilitv of complete bipartite graphs. The Electronic Journal of Combinatorics. 2002. V. 9, N 9. P. 1.
9. Raigorodskii A.M., Cherkashin D.D. Extremal problems in hvpergraph colourings. Russian Mathematical Surveys. 2020. V. 75, N 1. P. 89-146.
10. Schauz U. A paintabilitv version of the combinatorial Nullstellensatz, and list colorings of k k
11. Thomassen C. The even cycle problem for directed graphs. Journal of the American Mathematical Society. 1992. V. 5, N 2. P. 217-229.
12. Vizing V.G. Vertex colorings with given colors. Diskret. Analiz. 1976. V. 29. P. 3-10.
Поступим в редакцию 24-03.2022
