On-Line Choice Number of Complete Multipartite Graphs: an Algorithmic Approach

This paper studies the on-line choice number on complete multipartite graphs with independence number $m$. We give a unified strategy for every prescribed $m$. Our main result leads to several interesting consequences comparable to known results. (1) If $ k_1-\sum_{p=2}^m(\frac{p^2}{2}-\frac{3p}{2}+1)k_p\geq 0$, where $k_p$ denotes the number of parts of cardinality $p$, then $G$ is on-line chromatic-choosable. (2) If $ |V(G)|\leq\frac{m^2-m+2}{m^2-3m+4}\chi(G)$, then $G$ is on-line chromatic-choosable. (3) The on-line choice number of regular complete multipartite graphs $K_{m\star k}$ is at most $(m+\frac{1}{2}-\sqrt{2m-2})k$ for $m\geq 3$.


Definition 1 Given a graph G and an integer-valued function f on V (G), the on-line
f -list coloring of G is a two-players game, say Alice and Bob, played on G. In the very beginning, all vertices are uncolored. In the ith round, Alice marks a nonempty subset V i of remaining uncolored vertices and assigns color i as a permissible color to each vertex of V i . Then Bob chooses an independent set X i contained in V i and colors all vertices of X i the color i. The game goes round by round. If at the end of some round there is a vertex v which has been assigned f (v) permissible colors, i.e., has been marked f (v) times, but is not yet colored by Bob, then Alice wins the game.
Otherwise, Bob wins, i.e., in the end each vertex v is colored by Bob before running out of f (v) permissible colors.
Given an integer-valued function f defined on V (G), we say that G is on-line f -choosable if Bob has a winning strategy for the on-line f -list coloring game on G no matter how Alice plays; particularly, if f (v) = k, a constant, for all v ∈ V (G) then we say that G is on-line k-choosable. Denoted by χ p (G), the on-line choice number of G is the minimum number k such that G is on-line k-choosable.
The conventional list coloring, introduced by Vizing [14] and independently by Erdős, Rubin and Taylor [3], is a special case that Alice shows Bob the full lists in the very beginning of the on-line list coloring game. So Bob has a winning strategy for the list coloring if he has one for the on-line list coloring game. Let χ(G) and χ ℓ (G) denote the chromatic number and choice number of a graph G, respectively. In general, we have χ(G) ≤ χ ℓ (G) ≤ χ p (G) for any G.
It is known that χ ℓ (G) − χ(G) can be arbitrarily large; see [5] for an example that demonstrates complete bipartite graphs G having χ ℓ (G) arbitrarily large but χ(G) = 2. An interesting question is that whether χ p (G) − χ ℓ (G) can be arbitrarily large. To the best of our knowledge, the problem is still open. Although there exist a few graphs G with χ p (G) > χ ℓ (G) (see [15]), the difference is 1.
We remark that to prove the on-line Ohba's conjecture, it suffices to prove it for complete χ(G)-partite graphs G.
Using the Combinatorial Nullstellensatz, the authors [6] proved that K 2⋆k is online k-choosable. Recently, Kim et al [8] gave an algorithmic proof for K 2⋆k and later Kozik, Micek and Zhu [10] extended to complete multipartite graphs with independence number at most 3. This paper focuses on complete multipartite graphs with independence number m. In Section 2, we generalize the algorithmic methods in [8,10] and give a unified strategy for the on-line choice number of graphs with any prescribed m. Our main result provides a sufficient condition on f for graphs being on-line f -choosable by partitioning vertices in a systematic way into independent sets. It is a broadly applicable tool which leads to several interesting consequences comparable to known results. Section 3 presents some immediate consequences.

Main Result
Consider a complete multi-partite graph G with part size at most m. Let Π = {X m−1 , X m−2 , · · · , X 1 , X 2 , · · · , X m } be a partition of parts of G such that X p contains only parts of size p for 2 ≤ p ≤ m and X p contains only parts of size at most p for 1 ≤ p ≤ m − 1 (particularly, X 1 contains only parts of size 1). Let u p and ℓ p denote the number of parts of X p and X p , respectively, i.e., |X p | = u p and |X p | = ℓ p . For each family X p , we give an ordering on its parts and shall use subscripts to label them, e.g., X p = (X p 1 , · · · , X p ℓp ) where X p i means the ith part in X p . When it comes to vertices in a family of parts, we shall use the notation V (·) to avoid confusion.
The following are elementary but useful observations.
Proof. The proof follows immediately by definition.
Then for any integers s and t with 1 Proof. The proof follows from the expressions of α(j) and β(j).
Proposition 3 For j ≥ 2, every coefficient of u p 's in α(j) is at least max{j, p}.
Throughout the paper, U shall be used to denote the set Alice marks and I ⊆ U denotes the set Bob removes. For any U ⊆ V (G), the indicator function 1 U of U is Proposition 4 [10,13] If G is edgeless and f (v) ≥ 1 for all v ∈ V (G), then G is on- Theorem 2.1 Let G be a complete multi-partite graph with independence number m ≥ 2. If there are a partition Π of parts of G as described and a function f : V (G) → N satisfying the following: Proof. We shall prove the theorem by induction on by (R1) and f (v) ≥ 1 for v ∈ V ( m−1 p=1 X p ) by (R2). Assume that G has at least two parts and that the statement is true for all graphs of order less than |V (G)|. We shall prove that if G has a partition Π of parts and a function f : V (G) → N so that (R1) and (R2) are satisfied, then no matter what U ⊆ V (G) Alice marks, there exists an independent set I ⊆ U of G such that the resulting graph G ′ = G − I satisfies the two conditions, i.e., there exists a partition Π ′ of parts of G ′ such that f ′ = f − 1 U with respect to Π ′ satisfies (R1) and (R2). Then by induction we conclude that G ′ is on-line f ′ -choosable and thus G is on-line f -choosable by Proposition 4.
For a given U ⊆ V (G), the crucial step is twofold: decide an independent set I ⊆ U and rebuild a partition Notice that our strategy will be given case by case depending on U. Particularly, in any considered case we shall assume that all the previous cases do not hold. Note that from Π to Π ′ all families are inherited except two: the family from which I is chosen and the family where the remaining partite set X −I is inserted. We comment the only two families and orderings therein if necessary. We shall use the notations (X j , X − I) and (X − I, X j ) to denote that the remaining set X − I is inserted to the end and the beginning of the family X j , respectively. Note also that, once U is given, the function f ′ can be obtained from f with little difference 1 U . So we may and shall verify the inequalities in (R1) and (R2) for f ′ and Π ′ by comparing the difference with that for f and Π.
where the last inequality follows from This follows immediately from u ′ j * = u j * − 1 and Proposition 1.
Among all those cases we choose the one with the largest y. Let I = U ∩ X and Π ′ = Of particular note is that j * > y for otherwise it is Case 1.
(R1). Consider F ′ (j) for the case j ≤ y, which implies j * > j. Since j * > j and Since there are j * − y * elements inserted to the family X ′ m−y and m − y ≤ m − j, by . We now verify (R1), where the first inequality holds by the maximality assumption of y.
If j < j * , then where the last inequality can be proved through two cases: If y * ≥ j, then it follows from the same analysis as the previous case; if y * < j, then it follows from the fact When j * = 3, either y = 1 or y = 2. This implies the term y(j * − y) in the above where the last inequality follows from u ′ j * = u j * − 1, Proposition 1 and the fact that the last two terms are inherited.
For the case j = m − y and i = ℓ m−y + 1, i.e., v ∈ X − I. In this case, For the case j > m − y, we have j * > y * ≥ y ≥ m − j + 1. Therefore, by as desired.
Note that as Cases 1 and 2 were excluded, in all remaining cases we conclude that if U ∩ X = ∅ for some X ∈ X j , then (i) |U ∩ X| < j for otherwise it is Case 1, (ii) for all subsets Y ⊆ U ∩ X, F (|Y |) is not saturated with respect to Y for otherwise it is Case 2.
Case 3: From the above discussion, we know that one of the following cases must (R1). Consider any subset J ⊆ X ∈ X j * with |J| = j for some j * . Notice that j ≤ j * . Here we may assume that If F (j) is saturated with respect to J, then J U (by (ii)) and j * ≤ s (for and the coefficient of |V (X m+1−s )| in the expression of β(j) is j − 1 (from j ≤ s and Proposition 2), we have β(j) ≥ β ′ (j) + (j − 1). It follows that If F (j) is not saturated with respect to J and J ⊆ U, then j ≤ s (for otherwise it is where the first inequality holds as F (j) is not saturated with respect to J.
(R2). Because of the maximality of s, it suffices to consider v ∈ V (X ′ j i ) for the case j > m + 1 − s and the case j = m + 1 − s and i > i * as f ′ (v) is inherited otherwise. If q )|, as desired. For the case that i > i * and j = m + 1 − s, because of the minimality of i * , we Case (3.2): Note that in this case, t is the largest, i.e., t > s and t ≥ u. We such that F (t) is not saturated with respect to Y . Let I = U ∩ X (noticing that j * > |I| ≥ t) and Π ′ = Π − I where all families stay put except that Observing the corresponding coefficients of u j * , u j * −|I| and |V (X 1 )| in the expression of α(j) + β(j), we have that a j * − a j * −|I| and a j * − b 1 are at least |I| by Proposition 2. Accordingly, we can conclude that α(j) + β(j) ≥ (R1). Consider any subset J ⊆ K ∈ X k with |J| = j for some k and J ∩ U = ∅.
Notice that j ≤ k.
Consider the case that k ≤ t and F (j) is saturated with respect to J, which implies J U by (ii). It follows that w∈J f ′ (w) ≥ α(j) + β(j) − (j − 1). Since If F (j) is saturated with respect to J and k > t, then J ∩ U must be empty .
If F (j) is not saturated with respect to J, then j ≤ t for otherwise it is Case (3.2) with j > t violating the maximality assumption of t. It follows that w∈J f ′ (w) ≥ From the above discussion, in either case we have w∈J f ′ (w) ≥ α ′ (j) + β ′ (j). As J is chosen arbitrarily, we can conclude that If it is the case (a), then S ′ (m − j) ≤ S(m − j) (as s j * ≥ s j * −|I| by Proposition 1) q )| + 1, as desired. If j > m + 1 − t, then m + 1 − j < j * since t < j * for otherwise it is Case 1. In this case, observing the coefficients of u j * and u j * −|I| in S(m − j), we obtain s j * > s j * −|I| and s j * ≥ 2 by Proposition 1.
If it is the case (a), then S(m − j) ≥ S ′ (m − j) + 1 (since s j * > s j * −|I| ) and thus and i−1 q=1 |V (X ′ j q )| are inherited. If it is the case (b), then S(m − j) ≥ S ′ (m − j) + 2 (since s j * ≥ 2 and u ′ j * = u j * − 1) and j−1 Case (3.3): Note that in this case, u is the largest, i.e., u > s and u > t.
To avoid confusion, instead of u we shall use j * to denote the largest index and assume that there exists Y U and Y ⊆ X ∈ X j * such that F (|Y |) is saturated with respect to Y . Among all these cases with the same j * , we choose the one with the largest |U ∩ X|. Let I = U ∩ X and Π ′ = Π − I where all families stay put The same argument as that in (R1). Consider any subset J ⊆ K ∈ X k with |J| = j for some k and J ∩ U = ∅.
Notice that j ≤ k.
If F (j) is saturated with respect to J, then k ≤ j * for otherwise it is Case (3.3) with k > j * violating the maximality assumption of j * . Next we discuss the two cases k = j * and k < j * solely.
where the first inequality follows from the fact that among all the cases with k = j * we choose the largest |U ∩ X|.
If j ≥ j * , then we have J U for otherwise it is Case (3.2) with t = j ≥ j * = u.
Furthermore, it must be |J ∩ U| < j * for otherwise either F (|J ∩ U|) is saturated with respect to J ∩ U (Case 2) or F (|J ∩ U|) is not saturated with respect to J ∩ U and t = |J ∩ U| ≥ j * = u (Case (3.2)). Since F (j) is not saturated with respect to J and |J ∩ U| < j * , we have w∈J f ′ (w) ≥ α(j) + β(j) + 1 − (j * − 1). Since From the above discussion, in either case we have w∈J f ′ (w) ≥ α ′ (j) + β ′ (j). As J is chosen arbitrarily, we can conclude that |V (X j q )| + 1. If it is the case (a), then S ′ (m − j) ≤ S(m − j) (by Proposition 1 and j * ≥ j * − |I|) If it is the case (a), then to verify f ′ (v) it suffices to compare the coefficients of u j * and u j * −|I| in S(m − j). Since j * > m + 1 − j, we have s j * > s j * −|I| by Proposition 1.
. As one of the cases discussed above must occur, by induction the proof is complete.
Consider any subset J ⊆ X ∈ m p=2 X p with |J| = j, 1 ≤ j ≤ m. To verify (R1), it suffices to prove that w∈J f (w) ≥ α(j) + β(j), or equivalently that Obviously, Eq.(4) is always true for j = 1. By using elementary calculus, it is easy to prove that for any j = 2, 3, · · · , m with j ≤ p−1, This is trivially true as For any two graphs G and H, denote by G H the join of G and H, that is, the disjoint union of G and H with the edges {uv : u ∈ V (G), v ∈ V (H)}. Kozik, Micek and Zhu [10] proved that for any graph G, the join of G and a complete graph of order |V (G)| 2 is on-line chromatic-choosable. Later, Carraher, Loeb, Mahoney, Puleo, Tsai and West [2] improved upon |V (G)| 2 with an additional assumption. Precisely, they proved that for every d-degenerate graph G having an optimal proper coloring with color classes of size at most m, G K t is on-line chromatic-choosable if t ≥ (m + 1)d.  Kozik, Micek and Zhu [10] commented that when |V (G)| ≤ χ(G) + χ(G), G is on-line chromatic-choosable. Later, Carraher et al. [2] showed that the same conclusion holds under a relaxed condition |V (G)| ≤ χ(G) + 2 χ(G) − 1. They proposed a weak version of Conjecture 1: Conjecture 2 (Weak On-Line Ohba's Conjecture) [2] There is a constant c ∈ (1, 2] such that χ p (G) = χ(G) whenever |V (G)| ≤ cχ(G).
The weak on-line Ohba's conjecture is still open, to the best of our knowledge. Following the same argument in Theorem 3.1, we obtain the following result, which goes one further step towards the weak conjecture.
Alon [1] established the asymptotically tight bound χ ℓ (K m⋆k ) = Θ(k log m). The following result, which is another immediate consequence of Theorem 2.1, gives a general upper bound for χ p (K m⋆k ).