Forbidden Bicliques Bound Tree-Independence in P5-Free Graphs
A decomposition-repair proof establishes a $4\ell-4$ bag-independence bound when induced $P_5$ paths and $K_{\ell,\ell}$ bicliques are excluded.
Sep 23, 20264 min2605.03965
Combinatorics
math.CODiscrete mathematics, graph theory, enumeration, and optimization.
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Three Obstructions Characterize Finite Path-Width in Countable Graphs
Structural exclusions yield a finite path-width criterion, while universal-host constructions separate line-width bounds from path-width guarantees.
Sep 5, 20264 min2608.27752
Intersective Polynomial Difference-Free Sets Receive Stronger Density Bounds
Arithmetic level-d estimates, sieve-based exponential sums, and random sparsification yield a stretched-exponential upper bound for every exponent below 1/2.
Aug 31, 20263 min2605.16216
Hypergraph Alon-Tarsi Bounds Track Edge Density
Permuting nonconstant edge coefficients gives an arbitrary-field bound of $2\lceil\mathrm{ed}(H)\rceil+1$, connecting polynomial structure to hypergraph density.
Aug 21, 20264 min2501.00157
Two Rainbow Cycles Extend Lehel’s Partition Theorem
Under proper edge colouring, an existence proof partitions every sufficiently large complete graph into two vertex-disjoint rainbow cycles.
Aug 19, 20264 min2608.17996
Universal Density Parameter Determines Graph Percolation Thresholds
Unfolding and refolding witness graphs lets the authors characterize critical activation density for every fixed graph H, including witnesses larger than the host graph.
Aug 14, 20264 min2605.15066
Non-Backtracking Walks Reach Collisions Near the Square-Root Scale
Expected self-intersection time is O(√n log n) on bounded-degree graphs and O(√n) for regular graphs with a uniform spectral gap.
Aug 11, 20263 min2608.09729
Comparability-Graph Unions Can Have Chromatic Number k^d
For every positive d and k, the authors construct a union of d comparability graphs with clique number k and chromatic number k^d, extending the separation to fractional coloring and the independence ratio.
Aug 9, 20262 min2606.09415
Reflection Labels Establish Shellability for Shifted Lower Bruhat Intervals
An explicit reflection labeling gives every shifted lower interval a controlled chain structure across arbitrary Coxeter groups.
Aug 6, 20264 min2608.04417
Spectral Certificates Separate Accuracy From Inference
An explicit Matrix-Bernstein and gap protocol gives finite-sample network certificates, then declines when valid sets are still vacuous.
Jul 30, 20265 min2602.10566