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gpt 6 astra ultra attempt on Math-1107

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. List chromatic number and maximum degree of bipartite graphs
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Scope and process.
  6. Origin, literature and plan.
  7. A palette-splitting lemma with its quantifiers checked.
  8. Explicit examples requiring logarithmically many list colours.
  9. Why the order dependence is a real gap in this proof.
  10. Final status.
  11. Sources

List chromatic number and maximum degree of bipartite graphs

Definitions and mathematical statement

Let $G=(V,E)$ be a finite simple bipartite graph: there is a partition $V=A\sqcup B$ such that every edge has one end in each part. Let $\Delta(G)$ be the largest vertex degree. A graph is $k$-choosable if, for every assignment of a $k$-element set $L(v)$ of colours to each vertex, there is a proper colouring $c$ with $c(v)\in L(v)$. The least such $k$ is the list chromatic number $\chi_\ell(G)$; for a nonempty edgeless graph it equals one.

The logarithmic-bound conjecture asks whether there exists an absolute constant $C>0$ such that, for every finite bipartite graph with $\Delta(G)\ge2$, \[ \chi_\ell(G)\le C\log\Delta(G). \] Here $\log$ denotes the natural logarithm. Changing its base merely changes $C$. The restriction $\Delta\ge2$ makes the formula meaningful: $\log1=0$ cannot bound the choice number of an edge. An equivalent all-degree formulation is that some constant $C'>0$ satisfies $\chi_\ell(G)\le C'\log(\Delta(G)+2)$ for every nonempty bipartite graph.

The same constant must work independently of the order, the relative sizes of the two parts, and all overlaps among lists. Bipartite graphs have ordinary chromatic number at most two, but that does not give a two-colour list assignment compatible with arbitrary vertex lists. The target concerns this additional list-colouring difficulty rather than ordinary two-colourability.

Short English statement

Is the list chromatic number of every bipartite graph bounded by a constant times the logarithm of its maximum degree?

Research attempt

Scope and process.
This is a GPT-6 Astra Ultra research attempt dated 2 October 2026, using primary-source browsing and elementary mathematical checks. The canonical definition above is retained. Results below are restricted results or reductions, not a claim of a new solution to the entire catalogue question. No formal proof assistant or external mathematical referee checked this text.
Origin, literature and plan.
This is the Alon--Krivelevich logarithmic maximum-degree conjecture: the original paper [R1] states it as Conjecture 5.1. It is distinct from the older minimum-order problem for nonchoosable bipartite graphs. Bradshaw, Mohar and Stacho [R2] give a general upper bound on the scale $\Delta/\log\Delta$, which does not supply the logarithmic scale requested here. We prove the target for components of order polynomial in $\Delta$, give explicit logarithmic lower-bound examples, and identify the extra dependence that blocks the simple probability argument in unrestricted graphs.
A palette-splitting lemma with its quantifiers checked.
Let $G$ be bipartite with classes $X,Y$ and $N$ vertices, and assign a list $L(v)$ of $q$ colours to each vertex. Independently send every colour name appearing in any list to one of two palettes, each with probability $1/2$. A vertex of $X$ is bad if its list has no colour in the first palette; a vertex of $Y$ is bad if its list has none in the second. Each individual bad event has probability $2^{-q}$, regardless of list overlaps. A union bound gives probability at most $N2^{-q}$ that any vertex is bad.

Whenever $N2^{-q}<1$, there exists a split for which every vertex has an allowed colour in its designated palette. Choose any such colour at each vertex. Edges cross the bipartition and the palettes are disjoint, so the colouring is proper. This proves \[ \chi_\ell(G)\le\lfloor\log_2N\rfloor+1. \] Strict inequality in the probability bound matters when $N$ is a power of two; the displayed extra one handles that case.

Suppose every connected component has at most $\Delta^a$ vertices for some fixed $a>0$ and $\Delta\ge2$. Apply the lemma separately to components. Even if colour names occur in different components, the palette choices may differ because there are no cross-component edges. We obtain \[ \chi_\ell(G)\le a\log_2\Delta+1 \le\frac{a+1}{\log2}\log\Delta. \] This verifies the conjectured form on that family, with a constant depending only on the fixed exponent $a$. In particular it applies to balanced complete bipartite graphs, whose order is twice their maximum degree, by the original $N$-bound.

Explicit examples requiring logarithmically many list colours.
Fix $q\ge2$, take a colour universe of size $2q-1$, and put $n=\binom{2q-1}{q}$. On each side of $K_{n,n}$ assign every $q$-element subset of that universe once as a list. In any colouring, the colours used on one side must meet every $q$-subset. Such a hitting set has at least $q$ elements, since a set of at most $q-1$ elements has a complement containing a $q$-subset. Both sides thus use at least $q$ colours. The total universe has only $2q-1$, so the two used-colour sets intersect. Every cross-pair is adjacent, contradicting properness. Therefore $\chi_\ell(K_{n,n})>q$.

Since $n\le2^{2q-1}<4^q$, this family gives a lower bound of logarithmic order in $\Delta=n$. It does not contradict the target; it proves that a bounded constant independent of degree could not replace its logarithm. For $q=2$ the construction is $K_{3,3}$ with lists $\{1,2\},\{1,3\},\{2,3\}$ on each side. The companion script [R3] checks noncolourability for $q=2,3$ by enumerating all partitions of the finite colour universe between the sides, rather than assuming independent vertex choices.

Why the order dependence is a real gap in this proof.
A connected bipartite graph can have fixed maximum degree and arbitrarily many vertices: paths already do. Thus $\log N$ cannot be substituted by $O(\log\Delta)$ in the unrestricted palette lemma. Paths themselves are two-choosable by colouring from an endpoint, so they are not counterexamples; they expose a weakness of the method. Likewise a colour may appear in lists at arbitrarily many far-apart vertices. The dependency graph of palette-failure events is therefore not bounded by $\Delta$ merely because the graph has bounded degree. Invoking a local lemma without an additional dependency analysis would not remove the unwanted order factor.
Final status.
UNRESOLVED. The order-controlled upper bound and explicit lower-bound family are proved. The missing step is a colouring argument whose cost depends on graph degree while tolerating arbitrarily large components and unrestricted global overlaps among lists.

Sources