A Paper a Week

Current Schedule

WEEK 1 August 3–7, 2026

On the Gracesize of Trees

S. Letzter, A. Pokrovskiy, and E. Williams

arXiv, 2025 View paper ↗
WEEK 2 August 10–14, 2026

Orientations of Graphs Avoiding Given Lists on Out-Degrees

S. Akbari, M. Dalirrooyfard, K. Ehsani, K. Ozeki, and R. Sherkati

Journal of Graph Theory, 2020 View paper ↗
WEEK 3 August 17–21, 2026

Circuits through prescribed edges

P. Knappe and M. Pitz

Journal of Graph Theory, 2020 View paper ↗
WEEK 4 August 24–28, 2026

Line Transversals in Families of Connected Sets in the Plane

D. McGinnis and S. Zerbib

SIAM Journal on Discrete Mathematics, 2022 View paper ↗

Future Papers

Short Paper Bank

  • An Improved Upper Bound for the Multicolour Ramsey Number of Odd Cycles — M. Axenovich, W. Cames van Batenburg, O. Janzer, L. Michel, and M. Rundström — Journal of Combinatorial Theory, Series B, 2026 — paper ↗
  • A note on tree-cycle Ramsey numbers — T. Huang, Y. Zhang, and Y. Chen — arXiv, 2026 — arXiv ↗
  • List 4-colouring of planar graphs — X. Zhu — Journal of Combinatorial Theory, Series B, 2023 — paper ↗
  • The Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs — Z.-X. Song — arXiv, 2026 — arXiv ↗

Paper Bank

  • Longest Path and Cycle Transversals in Chordal Graphs — J. A. Long, Jr., K. G. Milans, and M. C. Wigal — arXiv, 2024 — arXiv ↗
  • One or Two Disjoint Circuits Cover Independent Edges: Lovász–Woodall Conjecture — K. Kawarabayashi — Journal of Combinatorial Theory, Series B, 2002 — paper ↗
  • An improved upper bound for the planar Turán number of $C_8$ — X. Bai, W. Liu, X. Nie, and X. Zhang — arXiv, 2026 — arXiv ↗
  • Density Conditions For Triangles In Multipartite Graphs — A. Bondy, J. Shen, S. Thomassé, and C. Thomassen — Combinatorica, 2006 — paper ↗

Completed Papers

On the Gracesize of Trees

S. Letzter, A. Pokrovskiy, and E. Williams

arXiv, 2025 View paper ↗

BIGGEST TAKEAWAY

My biggest takeaway was a useful tree-decomposition lemma. For every sufficiently large tree \(T\) and every prescribed set $S\subseteq V(T)$ of bounded size, one can remove a small set \(W\subseteq V(T)\setminus S\) such that $T-(S\cup W)$ is a disjoint union of copies of a rooted forest $F$. Moreover, every root has at most one edge to $S$, while no non-root vertex has an edge to $S$.

Another takeaways is the symbiosis between deterministic constructions and randomness. We begin with a large universe \(U\) and seek an object containing a desired structure \(S\). We identify a tractable property \(P\) whose presence guarantees \(S\). We then construct a structured subuniverse \(C\subseteq U\) with a robust precursor property \(P_0\). Rather than sampling from all of \(U\), we take a random restriction, sample, or perturbation \(R\) of \(C\). Expectation calculations show that the deterministic structure \(P_0\) should transform into \(P\) inside \(R\). Finally, concentration shows that \(P\) holds with positive or high probability, and therefore some \(R\) contains \(S\). In the graceful-tree application, \(U\) is all possible triples consisting of two endpoint labels and their difference, while \(C\subseteq U\) is a deterministic reservoir obtained by retaining only carefully chosen packets of such triples. The precursor property \(P_0\) is that this reservoir is linear, has tightly controlled degrees, and has only a small exceptional boundary.3 The resulting property \(P\) is the near-regularity required by the hypergraph matching theorem, and the target structure \(\mathcal S\) is a large matching that simultaneously gives distinct vertex labels and distinct edge differences.

Orientations of Graphs Avoiding Given Lists on Out-Degrees

S. Akbari, M. Dalirrooyfard, K. Ehsani, K. Ozeki, and R. Sherkati

Journal of Graph Theory, 2020 View paper ↗

BIGGEST TAKEAWAY

My first takeaway is a lemma that says: if a bipartite graph $G$, with forbidden list assignment $F$, has an orientation $D$ satisfying $d_D^+(v)\geq |F(v)|$ for every vertex $v$, then $G$ has an $F$-avoiding orientation. The orientation $D$ is not itself expected to avoid the forbidden lists, instead it acts as a certificate that the forbidden sets can be reduced one edge at a time without losing the hypothesis needed for induction. To show this, if no set $F(v)$ contains either $0$ or $d_G(v)$, then the bipartition gives an immediate solution: orient every edge from one part to the other, so every outdegree is either $0$ or $d_G(v)$. Otherwise, one chooses a vertex whose forbidden set contains an extreme value and deletes an appropriately directed edge. The broader lesson is that an auxiliary orientation can keep track of how much room remains for induction, while the reduction rules show how to update the forbidden sets when an edge is deleted and later added back.

A second takeaway is the usefulness of a large spanning bipartite subgraph from an arbitrary graph. Every graph $G$ has a bipartition $(A,B)$ for which the spanning bipartite subgraph $H:=G[A,B]$ satisfies $d_H(v)\geq \frac{1}{2}d_G(v)$ for every $v\in V(G)$. Indeed, choose a bipartition maximizing the number of crossing edges. If some vertex $v$ had more neighbors on its own side than on the opposite side, then moving $v$ across the partition would increase the number of crossing edges, contradicting maximality. If we know something about bipartite graphs, this can be a way to achieve some results for non-bipartite graphs.