Graphs and CombinatoricsDOI 10.1007/s00373-014-1438-9

ORIGINAL PAPER

On Improperly Chromatic-Choosable Graphs

Zhidan Yan · Wei Wang · Nini Xue

Received: 2 April 2013 / Revised: 10 March 2014© Springer Japan 2014

Abstract A graph is d-improperly chromatic-choosable if its d-improper choice num-ber coincides with its d-improper chromatic number. For fixed d ≥ 0, we show thatif the d-improper chromatic number is close enough to 1

d+1 of the number of verticesin G, then G is d-improperly chromatic-choosable. As a consequence, we show thatthe join G + Kn is d-improperly chromatic-choosable when n ≥ (|V (G)| + d)2. Wealso raise a conjecture on d-improper chromatic-choosability.

Keywords Improper coloring · Improper choice number · Improper chromaticnumber

Mathematics Subject Classification (2000) MSC 05C15

1 Introduction

All graphs considered in this paper are finite, undirected and without loops or multipleedges. A coloring is a mapping f : V (G) → S, where S is a set of colors. We call theset f −1(c) = {v ∈ V (G) : f (v) = c} a color class for each c ∈ S. If |S| = k then wecall c a k-coloring. If each color class induces a subgraph of maximum degree at mostd, then we call f a d-improper coloring. We say G is d-improperly k-colorable if thereexists a d-improper k-coloring of G. The d-improper chromatic number χd(G) of G

Z. Yan (B) · W. Wang · N. XueCollege of Information Engineering, Tarim University, Alar 843300, Chinae-mail: yanzhidan.math@gmail.com

W. Wange-mail: wangwei.math@gmail.com

N. Xuee-mail: xuenini19222@163.com

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is the least k such that G is d-improperly k-colorable. Since a 0-improper k-coloringis a proper coloring, χ0(G) is the usual chromatic number of the graph. The conceptof improper colorings was introduced independently by several authors [1,2].

A list assignment of a graph G is a function L defined on V (G) such that L(v) is thelist of colors available for each v ∈ V (G). For a given positive integer k, if |L(v)| = kfor each v ∈ V (G), we say L is a k-list assignment of G. For a list assignment L , acoloring f is called an L-coloring if f (v) ∈ L(v) for each v ∈ V (G). We say G isd-improperly k-choosable if for any k-list assignment L of G there is a d-improperL-coloring of G. The d-improper choice number chd(G) (or d-improper list chromaticnumber) of G is the least k such that G is d-improperly k-choosable. The investigationof d-improper list colorings was launched by Eaton and Hull [4] and Škrekovski [13].Note that 0-improper choice number is the same as the usual choice number (a conceptintroduced by Vizing [14] and independently by Erdos et al. [6]).

A graph G has a d-improper k-coloring precisely if it has a d-improper L-coloring,where L(v) = {1, 2, . . . , k} for all v ∈ V (G). Therefore, chd(G) ≥ χd(G). A graphG is d-improperly chromatic-choosable if χd(G) = chd(G). While much attentionhas been paid on the usual chromatic-choosability (corresponding to d = 0), few isknown for the case d > 0. A result of Lovász [10] implies that χd(G) ≤ ��(G)+1

d+1 �,

moreover chd(G) ≤ ��(G)+1d+1 � as Lovász’s proof extends easily to list coloring. It

follows that complete graphs are d-improperly chromatic-choosable for any d.Ohba [12] conjectured that G is chromatic-choosable when |V (G)| ≤ 2χ(G) + 1,

and this has been proved by Noel, Reed, and Wu [11]. In the original paper of Ohba[12], he proved that G is chromatic-choosable when |V (G)| ≤ χ(G) + √

2χ(G). Wemention that two Ohba-like conjectures have been proposed recently. In [15], Zhen andWu considered for which graphs G, the list point arboricity ρl(G) coincides with itspoint arboricity ρ(G), a problem of chromatic-choosability in the sense that each colorclass induces a forest. They conjectured that if |V (G)| ≤ 3ρ(G) then ρ(G) = ρl(G).They also proved that if |V (G)| ≤ 2ρ(G) + √

2ρ(G) − 1 then ρ(G) = ρl(G). In [7]the authors proposed the on-line version of Ohba’s Conjecture: if |V (G)| ≤ 2χ(G)

then G is on-line chromatic-choosable. As a support to the on-line version of Ohba’sConjecture, Kozik, Micek and Zhu [9] proved that if |V (G)| ≤ χ(G) + √

χ(G) thenG is on-line chromatic-choosable. In [3], the authors showed that the conclusion alsoholds under a weaker condition that |V (G)| ≤ χ(G) + √

2χ(G) − 1.The current paper is to show that the analogous results hold for improper colorings.

We shall use ideas from [12] and [15] to prove the following theorem.

Theorem 1 If |V (G)| ≤ (d+1)χd(G)+√(d + 1)χd(G)−d, then χd(G) = chd(G).

Remark 1 When d = 0, the result of Theorem 1 is weaker than the original result ofOhba [12].

For two graphs G and H with disjoint vertex sets, the join of G and H , denotedby G + H , is obtained from their union by adding edges joining every vertex of G toevery vertex of H .

Corollary 1 If n ≥ (|V (G)| + d)2 then χd(G + Kn) = chd(G + Kn).

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Conjecture 1 If |V (G)| ≤ (d + 2)χd(G) + (d + 1) then χd(G) = chd(G).

Remark 2 Conjecture 1 agrees with Ohba’s conjecture when d = 0. It is well knownthat Ohba’s conjecture is best possible, as Enomoto et al. [5] proved that the choicenumber of the complete k-partite graph K4∗1,2∗(k−1) (with one partite set of size 4 andk − 1 partite sets of size 2) is equal to k + 1 if k is even. When d = 1 we shall showthat Conjecture 1 is best possible if it is true; see Corollary 4.

2 Proofs of Main Results

Let G be a graph with a list assignment L . Let X = {x1, x2, . . . , x p} ⊆ V (G). SetL(X) = L(x1) ∪ L(x2) ∪ · · · ∪ L(x p). Let G[X ] denote the subgraph of G inducedby X .

Lemma 1 If a graph G is not d-improperly L-colorable, then there exists a set X ⊆V (G) such that |X | > (d + 1)|L(X)|.Proof Suppose to the contrary that |X | ≤ (d + 1)|L(X)| for all subsets X of V (G).We consider the bipartite graph B with bipartition (V (G), C), where C consists of(d + 1) copies of L(V (G)) and each v ∈ V (G) is adjacent with (d + 1) copies ofL(v). Combining the definition of B with the assumption that |X | ≤ (d + 1)|L(X)|leads to |NB(X)| = (d + 1)|L(X)| ≥ |X | for X ⊆ V (G). Thus, by Hall’s Theoremthere is a matching M that saturates V (G). We color v by the color matched with it inM . Since C is (d +1) copies of L(V (G)), each element c ∈ L(V (G)) appears at most(d + 1) times as an end vertex of some edge of the matching M . That is, each colorc ∈ L(V (G)) is assigned to at most (d + 1) vertices of V (G) and hence each colorclass induces a subgraph of maximum degree at most d. Thus we get a d-improperL-coloring of V (G), a contradiction. � Lemma 2 Let G be a graph and let H be a graph with maximum degree at most d. IfG is d-improperly k-choosable and (|V (H)| − 1)(|V (G)| + |V (H)|) ≤ (d + 1)(k +1)|V (H)|, then G + H is d-improperly (k + 1)-choosable.

Proof Let L be a (k+1)-list assignment of G+H . We prove that G+H is d-improperlyL-colorable by induction on p = |V (H)|.

First let p = 1 and V (H) = {u}. We color u by one of the elements in L(u),say c. For any v ∈ V (G), let L ′(v) = L(v) \ {c}. Since |L ′(v)| ≥ |L(v)| − 1 ≥ kfor any v ∈ V (G), G is d-improperly L ′-colorable. Thus, G + H is d-improperlyL-colorable. �

Now we assume p ≥ 2 and let U = V (H).

Case 1. ∩u∈U L(u) �= ∅.Take a color c ∈ ∩u∈U L(u) and assign c to each vertex of H . For any v ∈ V (G),

let L ′(v) = L(v) \ {c}. Since L ′(v) ≥ |L(v)| − 1 ≥ k for any v ∈ V (G), G isd-improperly L ′-colorable. Thus, G + H is d-improperly L-colorable.

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Case 2. ∩u∈U L(u) = ∅.Suppose to the contrary that G + H has no d-improper L-coloring. By Lemma 1,

there must exist X ⊆ V (G + H) with |X | > (d +1)|L(X)|. We take X to be maximalin the sense of set inclusion.

Subcase 2.1. U ⊆ X .By the assumption of Case 2, each color in L(U ) appears in the lists of at most (p−1)

vertices of U and hence we have |L(X)| ≥ |L(U )| ≥ pp−1 (k + 1). On the other hand,

by the assumption of the lemma, we have |X | ≤ |V (G + H)| ≤ (d + 1)p

p−1 (k + 1).Thus we have |X | ≤ (d + 1)|L(X)|, a contradiction.Subcase 2.2. U \ X �= ∅.

Let Y = V (G + H) \ X and L ′(y) = L(y) \ L(X) for any y ∈ Y . If there is ad-improper L-coloring of (G + H)[X ] and a d-improper L ′-coloring of (G + H)[Y ],then we obtain a d-improper L-coloring of G + H . (This is a contradiction.)

First, we show that (G+H)[X ] has a d-improper L-coloring. Note that the functionf (p) = (1 − 1

p )(|V (G)| + p) is an increasing function and f (p) ≤ (d + 1)(k + 1)

by the assumption of the lemma, we have f (p − 1) ≤ (d + 1)(k + 1), i.e., (p −2)(|V (G)| + p − 1) ≤ (d + 1)(k + 1)(p − 1). Let u ∈ U \ X and H ′ = H − u. Byinduction hypothesis, G + H ′ is d-improperly (k + 1)-choosable. Since (G + H)[X ]is a subgraph of G + H ′, (G + H)[X ] has a d-improper L-coloring.

Now we show that (G + H)[Y ] has a d-improper L ′-coloring. Otherwise, there isa set S ⊆ Y with |S| > (d + 1)|L ′(S)| by Lemma 1. But, we have |L(X ∪ S)| =|L(X)|+ |L ′(S)| < 1

d+1 |X |+ 1d+1 |S| = 1

d+1 |X ∪ S|. This contradicts the maximalityof X . Thus, (G + H)[Y ] has a d-improper L ′-coloring. Now we can get a d-improperL-coloring of G + H .

Let K p1∗n1,...,pr ∗nr denote the complete multi-partite graph with ni partite sets ofsize pi . Erdoset al. [6] proved that K2∗n is chromatic-choosable. For n ≥ 1 and d ≥ 0,let us define K(d+2)∗n to be the class of graphs G that can be constructed by

G = F1 + F2 + · · · + Fn,

where F1, . . . , Fn be any pairwise vertex-disjoint forests with |V (Fi )| = d + 2 andδ(Fi ) = 1 for each i . If d = 0 then from the definition of Fi , each Fi is K2 and henceK(d+2)∗n is K2∗n . Since �(Fi ) = |V (Fi )| − 1 − δ(Fi ) = d, following [8], we mayregard any graph K ∈ K(d+2)∗n as a complete d-improperly n-partite graph. There-fore, χd(K ) ≤ n. We show that the equality holds and moreover K is d-improperlychromatic-choosable, a generalization of the fact that K2∗n is chromatic-choosable.

Corollary 2 χd(K ) = chd(K ) = n for any K ∈ K(d+2)∗n.

Proof We first show χd(K ) = n. Note χd(K ) ≤ n. Suppose to the contrary thatχd(K ) < n. Let f be a d-improper χd(K )-coloring of K and let S be the largestcolor class of f . Hence, |S| ≥ |V (K )|

χd (K )>

n(d+2)n = d + 2. We show �(K [S]) > d and

hence f is not d-improper, a contradiction.Let K = F1 + F2 + · · · + Fn and let Ui = V (Fi ) = V (Fi ) for i = 1, . . . , n.

Without loss of generality, we may assume S ∩ U1 �= ∅. Pick u ∈ S ∩ U1 such that

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dF1[S∩U1](u) = �(F1[S ∩ U1]). By the definition of graph join, it is clear that eachvertex in S \ U1 is a neighbor of u since u ∈ U1. Therefore,

dK [S](u) = dF1[S∩U1](u) + |S \ U1|= �(F1[S ∩ U1]) + |S \ U1|= (|S ∩ U1| − 1) − δ(F1[S ∩ U1]) + |S \ U1|≥ (|S ∩ U1| − 1) − 1 + |S \ U1|= |S| − 2

> d,

where the third equality use the fact that δ(G) + �(G) = |V (G)| − 1 for any graphG, and the fourth inequality follows from the fact that any subgraph of a forest is alsoa forest and hence has minimum degree at most one. Therefore, �(K [S]) > d, asdesired.

Next we show chd(K ) = n by induction on n. Since chd(K ) ≥ χd(K ) = n,it suffices to show chd(K ) ≤ n, i.e., K is d-improperly n-choosable. It triviallyholds when n = 1. Now we assume that the result holds for n. Let K = F1 +F2 + · · · + Fn+1 = G + H, where G = F1 + F2 + · · · + Fn and H = Fn+1. Byinduction hypothesis, G is d-improperly n-choosable. Since �(H) = d and (|V (H)|−1)(|V (G)| + |V (H)|) = (d + 2 − 1)(n + 1)(d + 2) = (d + 1)(n + 1)|V (H)|, weconclude that K = G + H is d-improperly (n + 1)-choosable by Lemma 2. � Corollary 3 χ1(K3∗n) = ch1(K3∗n) = n.

Proof It is clear that χ1(K3∗n) ≤ n. Since any induced subgraph of K3∗n with 4 ormore vertices has maximum degree at least 2, we see that χ1(K3∗n) ≥ n and henceχ1(K3∗n) = n. Note ch1(K3∗n) ≥ χ1(K3∗n). It remains to show ch1(K3∗n) ≤ n.

Let K ∈ K3∗n . Since K3∗n is a spanning subgraph of K , Corollary 2 impliesch1(K3∗n) ≤ ch1(K ) = n, as desired. �

The following corollary gives a family of graphs G with |V (G)| = 3χ1(G) + 3,none of which is 1-improperly chromatic-choosable.

Corollary 4 If n is even then ch1(K6∗1,3∗(n−1)) = χ1(K6∗1,3∗(n−1)) + 1 = n + 1.

Proof Clearly, χ1(K6∗1,3∗(n−1)) ≤ n. Since χ1(K6∗1,3∗(n−1)) ≥ χ1(K3∗n), Corollary3 implies χ1(K6∗1,3∗(n−1)) ≥ n. Therefore, χ1(K6∗1,3∗(n−1)) = n.

Next we show ch1(K6∗1,3∗(n−1)) = n + 1. Note that K6∗1,3∗(n−1) is a subgraphof K3∗(n+1). By Corollary 3, ch1(K6∗1,3∗(n−1)) ≤ n + 1. Therefore, it suffices toshow ch1(K6∗1,3∗(n−1)) > n. Denote the partite sets of K6∗1,3∗(n−1) by S1, S2, . . . , Sk

such that S1 = {x1, y1, x2, y2, x3, y3} and Si = {ui , vi , wi } for i ≥ 2. Let A, B, C bepairwise disjoint sets of colors with |A| = |B| = |C | = n

2 . Define an n-list assignmentL by

L(ui ) = A ∪ B, L(vi ) = B ∪ C and L(wi ) = C ∪ A for 2 ≤ i ≤ k, andL(x1) = L(y1) = A ∪ B, L(x2) = L(y2) = B ∪ C and L(x3) = L(y3) = C ∪ A.

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We show that K6∗1,3∗(n−1) is not 1-improperly L-colorable. Suppose to the contrarythat there is a 1-improper L-coloring f : V (K6∗1,3∗(n−1)) → A ∪ B ∪ C . Note thatL(ui ) ∩ L(vi ) ∩ L(wi ) = ∅ for 2 ≤ i ≤ n. For each c ∈ A ∪ B ∪ C such thatf −1(c) �⊆ S1, we have | f −1(c)| ≤ 2 since �(K6∗1,3∗(n−1)[( f −1(c)]) ≤ 1. Define abipartition (S, T ) of A ∪ B ∪ C as follows:

S = {c ∈ A ∪ B ∪ C : f −1(c) �⊆ S1} and T = {c ∈ A ∪ B ∪ C : f −1(c) ⊆ S1}.Let � = |∪c∈T f −1(c)|. Hence, |∪c∈S f −1(c)| = 3n+3−�, implying |S| ≥ ⌈ 3n+3−�

2

⌉

as | f −1(c)| ≤ 2 for each c ∈ S. Since any five vertices in S1 share no common color forthe list assignment L defined above, |T | ≥ ⌈

�4

⌉. As (S, T ) is a bipartition of A∪ B ∪C

and |A ∪ B ∪ C | = 3n2 , we have

⌈ 3n+3−�2

⌉ + ⌈�4

⌉ ≤ 3n2 , that is

⌈ 3−�2

⌉ + ⌈�4

⌉ ≤ 0. Adirect calculation indicates that this is a contradiction as 0 ≤ � ≤ |S1| = 6. �

Let G be a graph and let f : V (G) → {1, 2, . . . , χd(G)} be a d-improper χd(G)-coloring of G such that each color class contains at least d +1 vertices with at most oneexception. The requirement is satisfiable since any (d + 1)-subset of V (G) inducesa graph with maximum degree at most d. Let Vi = {v ∈ V (G) : f (v) = i} and|Vi | = ai for i = 1, . . . , χd(G). We assume a1 ≥ a2 ≥ · · · ≥ aχd (G) and ai ≥ d + 1for i = 1, . . . , χd(G) − 1.

Lemma 3 If(a1 − 1)|V (G)| ≤ (d + 1)a1χ

d(G), (1)

then χd(G) = chd(G).

Proof We prove this lemma by induction on χd(G). First, if χd(G) = 1 then �(G) ≤d and hence chd(G) = 1 = χd(G).

Now assume χd(G) ≥ 2 and thus a1 ≥ d +1. Since χd(G) ≤ chd(G), it suffices toshow that G is d-improperly χd(G)-choosable. If a1 = d + 1 then χd(G) ≥ �|V (G)|

d+1 �since a1 + · · · + aχd (G) = |V (G)| and a1 ≥ a2 ≥ · · · ≥ aχd (G). By Lovász’s bound,G is d-improperly χd(G)-choosable.

Now assume a1 ≥ d + 2. We claim (a2 − 1)|V (G − V1)| ≤ (d + 1)a2χd(G − V1).

We may assume a2 ≥ 2. By (1) we have

|V (G − V1)| = |V (G)| − a1

≤ (d + 1)a1

a1 − 1χd(G) − a1

= a1

a1 − 1((d + 1)χd(G) − (a1 − 1))

≤ a1

a1 − 1((d + 1)χd(G) − (d + 1))

≤ (d + 1)a2

a2 − 1(χd(G) − 1)

= (d + 1)a2

a2 − 1(χd(G − V1)).

The claim follows. By induction hypothesis, the graph G − V1 is d-improperly(χd(G) − 1)-choosable since V2 is the largest color class of G − V1. As G − V1

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is d-improperly (χd(G) − 1)-choosable and relation (1) holds, Lemma 2 implies thatG is d-improperly χd(G)-choosable. This completes the proof of this lemma. � Proof of Theorem 1. The theorem trivially holds if χd(G) = 1. We may assumeχd(G) ≥ 2. Since ai ≥ d + 1 for i = 2, . . . , χd(G) − 1 and aχd (G) ≥ 1, we havea1 ≤ |V (G)|−(d+1)(χd(G)−2)−1. Combining this inequality with the assumptionof this theorem, we have

a1 ≤ (d + 1)χd(G) +√

(d + 1)χd(G) − d − (d + 1)(χd(G) − 2) − 1

=√

(d + 1)χd(G) + d + 1.

Hence,

a1

a1 − 1(d + 1)χd(G) = (d + 1)χd(G) + (d + 1)χd(G)

a1 − 1

≥ (d + 1)χd(G) + (d + 1)χd(G)√

(d + 1)χd(G) + d

≥ (d + 1)χd(G) + (d + 1)χd(G) − d2√

(d + 1)χd(G) + d

= (d + 1)χd(G) +√

(d + 1)χd(G) − d

≥ |V (G)|.

Therefore, by Lemma 3, χd(G) = chd(G).

Proof of Corollary 1. Since χd(G + Kn) ≥ χd(Kn) = � nd+1� ≥ n

d+1 and n ≥(|V (G)| + d)2, we have

(d + 1)χd(G + Kn) +√

(d + 1)χd(G + Kn) − d ≥ n + √n − d

≥ n + (|V (G)| + d) − d

= |V (G + Kn)|.

Thus, Theorem 1 implies that χd(G + Kn) = chd(G + Kn). � Acknowledgments We thank the referees for their constructive comments. In particular, the current formof Conjecture 1 as a natural generalization of Ohba’s conjecture is inspired by one of the referees.

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