Shagnik Das, Nemanja Draganić, Raphael Steiner
Alon and Krivelevich proved that for every $n$-vertex subcubic graph $H$ and every integer $q \ge 2$ there exists a (smallest) integer $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which the length of every subdivision-path is divisible by $q$. Improving their superexponential bound, we show that $f(H,q) \le \frac{21}{2}qn+8n+14q$, which is optimal up to a constant multiplicative factor.
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We prove that for every graph $H$ of maximum degree at most $3$ and for every positive integer $q$ there is a finite $f=f(H,q)$ such that every $K_f$-minor contains a subdivision of $H$ in which every edge is replaced by a path whose length is divisible by $q$. For the case of cycles we show that for $f=O(q \log q)$ every $K_f$-minor contains a cycle of length divisible by $q$, and observe that this settles a recent problem of Friedman and the second author about cycles in (weakly) expanding graphs.
Divisible subdivisions of graphs in subdivisions of complete graphs
本文研究完全图边加权下图的q-可整除细分问题，改进了上界，并解决了q=2及树的情形，证明了精确值。关键贡献在于给出了更紧的界并解决了特定情形。
Nearly tight bounds for induced subdivisions
研究诱导细分问题，通过证明平均度为特定值时不含某些图的图包含\(K_h\)的诱导细分，给出了近乎紧界，改进了先前结果，且该界对\(h\)个顶点的其他图也适用。
The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term. We also show that if $T$ is an $n$-vertex tournament with $Δ^+(T)-δ^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is also tight up to multiplicative constant.
Nearly tight bound for rainbow clique subdivisions in properly edge-colored graphs and applications
本文证明了在正确边着色图中存在雨色K_t子图的近似界限，并应用于加性组合学、数论和编码理论。
Tight minimum degree conditions for apex-outerplanar minors and subdivisions in graphs and digraphs
研究最小度至少为|V(H)|-1的图必包含H作为子式的图类，证明一大类顶点外平面图满足此性质，并推广到有向图的细分和蝴蝶子式。
A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen sub…
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The …
We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\b…
A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $Ω(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.
The codegree Turán density of tight cycles
本文研究了k-均匀紧密循环的代码度Turán密度，得出了改进的上界和下界，并推广了之前的定理。
Crux, space constraints and subdivisions
本文引入 crux 度量图嵌入细分时的空间约束，建立由平均度与 crux 大小共同决定的最大团细分渐近最优界，并推出极值图刻画、二部禁子图结果及随机图二分性。
Induced subdivisions in $K_{s,s}$-free graphs with polynomial average degree
本文证明无 $K_{s,s}$ 图在多项式平均度条件下必含任意图 $H$ 的诱导细分，证实了 Bonamy 等人的猜想，并推广到不含 $C_4$ 的诱导子图及 $K_k$ 相关结构，回答了 Davies 的问题。
Improved upper bounds on longest-path and maximal subdivision transversals
本文改进了连通图Gallai数的上界，从$8n^{3/4}$降至$5n^{2/3}$，并推广到最大细分横贯问题，证明了更一般情形下的类似上界。
Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spann…
For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently large, then G can be tiled perfectly by copies of K_{h,h,h}. This extends the work in [19] and also gives a sufficient condition for tiling by any (fixed) 3-colorable graph. Furthermore, we show that this minimum-degree condition is best possible and provide very tight bounds when N is divisible by h but not by 6h.
A note on cube-free problems
本文研究了无立方体子集的大小上界问题，证明了在Z_N中2N/3的上界成立且当N被3整除时紧致，提出了d-无立方体子集的猜想并在最小素因子或素数幂等情形下证明，同时探讨了相关的无集合问题。
Semi-Degree Condition for Arbitrary $H$-Linked Oriented Graphs
本文提出任意H-连通有向图的半度数条件，证明了在特定度数下，有向图具有任意H-连通性。
Turán density of long tight cycle minus one hyperedge
本文研究删边紧圈 $\mathcal{C}^-_{\ell}$ 的 Turán 密度，借助锦标赛循环三角形的禁子超图表征，证明所有充分大且不被 3 整除的 $\ell$ 对应密度均为 $1/4$，并导出近似相似三角形上界的可检验证明。
Immersions of directed graphs in tournaments
本文研究锦标赛中有向图的浸入问题，证明了任意具有O(k)个顶点的锦标赛必含k个顶点的传递锦标赛的1-浸入，并进一步证明最小出度至少为Ck的锦标赛必含k个顶点的完全有向图的2-浸入，且这些界在相应意义下都是紧的。
Packing subgraphs in regular graphs
研究证明正则图中存在覆盖所有顶点的子图打包方法，解决了Kühn和Osthus提出的双图覆盖问题。
Openly disjoint cycles and directed tree-width of regular digraphs
研究正则有向图中公开不交环的数量及其与定向树宽的关系，证明正则有向图的公开不交环数下界与上界，推动有向图子图分解的度数条件研究。
Dominating $K_t$-Models
本文研究支配 $K_t$-模型，证明无支配 $K_t$-模型的图是 $2^{t-2}$-可染色的，并给出最大平均度上界 $2^{t-2}$ 与随机图下界，同时为强化哈德维格猜想提供证据。
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