On the dynamic width of the 3-colorability problem

A graph $G$ is 3-colorable if and only if it maps homomorphically to the complete 3-vertex graph $K_3$. The last condition can be checked by a $k$-consistency algorithm where the parameter $k$ has to be chosen large enough, dependent on $G$. Let $W(G)$ denote the minimum $k$ sufficient for this purpose. For a non-3-colorable graph $G$, $W(G)$ is equal to the minimum $k$ such that $G$ can be distinguished from $K_3$ in the $k$-variable existential-positive first-order logic. We define the dynamic width of the 3-colorability problem as the function $W(n)=max_G W(G)$, where the maximum is taken over all non-3-colorable $G$ with $n$ vertices. The assumption $mathrm{NP} emathrm{P}$ implies that $W(n)$ is unbounded. Indeed, a lower bound $W(n)=Omega(loglog n/logloglog n)$ follows unconditionally from the work of Nesetril and Zhu on bounded treewidth duality. The Exponential Time Hypothesis implies a much stronger bound $W(n)=Omega(n/log n)$ and indeed we unconditionally prove that $W(n)=Omega(n)$. In fact, an even stronger statement is true: A first-order sentence distinguishing any 3-colorable graph on $n$ vertices from any non-3-colorable graph on $n$ vertices must have $Omega(n)$ variables. On the other hand, we observe that $W(G)le 3,alpha(G)+1$ and $W(G)le n-alpha(G)+1$ for every non-3-colorable graph $G$ with $n$ vertices, where $alpha(G)$ denotes the independence number of $G$. This implies that $W(n)lefrac34,n+1$, improving on the trivial upper bound $W(n)le n$. We also show that $W(G)>frac1{16}, g(G)$ for every non-3-colorable graph $G$, where $g(G)$ denotes the girth of $G$. Finally, we consider the function $W(n)$ over planar graphs and prove that $W(n)=Theta(sqrt n)$ in the case.

Definability of linear equation systems over groups and rings

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.