Towards on-line Ohbas conjecture

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 \chi(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead’s result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le \chi(G)+\sqrt{\chi(G)}$ then its on-line choice number equals chromatic number.

💡 Research Summary

The paper investigates the online list‑colouring variant of the classic choice number problem. In the online setting a two‑player game is played: at each round the Lister presents a non‑empty set of still‑uncoloured vertices, and the Painter must select an independent subset to colour with the current colour. Each vertex v is equipped with a prescribed number f(v) of allowable colours; the game ends unfavourably for the Painter if some vertex runs out of permissible colours before being coloured. The online choice number ch_{OL}(G) is the smallest integer k for which the graph G is online k‑choosable (i.e., f(v)=k for all v). It is known that ch_{OL}(G)≥ch(G) for every graph, and strict inequality occurs for some families.

Ohba’s conjecture (now a theorem) states that if |V(G)|≤2χ(G)+1 then ch(G)=χ(G). An online analogue was proposed: if |V(G)|≤2χ(G) then ch_{OL}(G)=χ(G). This conjecture remains open in general. The authors settle it for the important class of graphs whose independence number α(G)≤3.

The technical core is Lemma 5, which gives a sufficient condition for a complete multipartite graph whose parts have size at most three to be online f‑choosable. The vertices are partitioned into four families: A (singletons), B (pairs), C (triples) and S (singletons or pairs). For a function f:V→ℕ the lemma requires a collection of inequalities (1)–(3.3) that relate f(v) to the numbers of parts of each type. The proof proceeds by induction on the number of vertices. At each step the Painter selects an independent set I⊆U (where U is the set presented by the Lister) according to a case analysis: if a whole triple is present, colour it; else if a whole pair is present, colour it; otherwise use a “deficit” argument. The deficit of a part measures how far the current colour budget f(v) is from the degree+1 bound; each move strictly reduces the deficit of every part of size at least two, and parts whose deficit reaches zero disappear. After at most |V(G)|/2 rounds the remaining graph is a clique, and the degree‑plus‑one condition guarantees that the Painter can finish the colouring. Hence the conditions of Lemma 5 are preserved under the removal of I, establishing the inductive step.

Using Lemma 5 the authors obtain two major consequences. First, they prove the online Ohba conjecture for all graphs with α(G)≤3: if |V(G)|≤2χ(G) then ch_{OL}(G)=χ(G). This settles the conjecture for a large and natural family of graphs. Second, they apply the lemma to the complete k‑partite graph K_{3⋆k} (k parts of size three). By setting f(v)=⌈3k/2⌉ they verify the lemma’s inequalities, yielding the bound ch_{OL}(K_{3⋆k})≤⌈3k/2⌉. Combined with Kierstead’s classic result ch(K_{3⋆k})=⌈(4k−1)/3⌉, this provides an alternative proof of the exact choice number and shows that the online choice number exceeds the ordinary choice number by at most a constant factor.

The paper also contains a structural result (Theorem 3) stating that for any graph G, adding a sufficiently large complete graph K_n (the join G+K_n) makes the resulting graph online chromatic‑choosable, i.e., χ(G+K_n)=ch_{OL}(G+K_n). By choosing n≈|V(G)|²/2 and analysing the deficit reduction process, the authors deduce Corollary 4: if |V(G)|≤χ(G)+√χ(G) then ch_{OL}(G)=χ(G). This extends the range of graphs for which the online choice number coincides with the chromatic number beyond the original Ohba bound.

Overall, the work introduces a novel “deficit‑reduction” technique that replaces the Hall‑matching arguments used in the classical proof of Ohba’s conjecture, which do not transfer to the online setting. The method is robust enough to handle multipartite graphs with parts of size three and yields both new upper bounds for online choice numbers and a complete verification of the online Ohba conjecture for graphs of independence number three. The results open avenues for further exploration of the gap between ch_{OL} and ch, and suggest that similar inductive, part‑wise strategies may be effective for broader classes of graphs.