不含$P_5$且无爪图中最长路径的非空交

Bi-traceable graphs, the intersection of three longest paths and Hippchen's conjecture

Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible configurations of the intersection points when $\#V(P)\cap V(Q)\le 5$ in order to prove that if the intersection of three longest paths $P,Q,R$ is empty, then $\#(V(P)\cap V(Q))\ge 6$. We also prove Hippchen's conjecture for $k\le 6$: If a graph $G$ is $k$-connected for $k\le 6$, and $P$ and $Q$ are longest paths in $G$, then $\#(V(P)\cap V(Q))\ge 6$.

具有路径约束的图上的警察与强盗问题

Cops and Robbers on Graphs with Path Constraints

本文研究了$P_k$-free图的cop number上界猜想，通过分析$(P_k,E)$-free图等结构，证明了新的cop number上界，改进并推广了现有结果。

无爪图中的 d-Cut 问题

Finding d-Cuts in Claw-free Graphs

该论文研究无爪图中的 d-Cut 问题，证明 d=2 时 NP-完全，并给出最大度限制下的复杂度划分：p≤2d+1 可常数时间求解，p≥2d+3 为 NP-完全，且可推广到 S_{1^t,l}-自由图。

Rich square-free words

A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantová and Starosta (Discrete Math. 313 (2013)) proved that every infinite rich word contains a square. We will give another proof for that result. Pelantová and Starosta denoted by r(n) the length of a longest rich square-free word on an alphabet of size n. The exact value of r(n) was left as an open question. We will give an upper and a lower bound for r(n), and make a conjecture that our lower bound is exact. We …

Extending Grundy domination to $k$-Grundy domination

The Grundy domination number of a graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$ satisfying $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$ for each $i \in [k]$. Recently, a generalization of this concept called $k$-Grundy domination was introduced. In $k$-Grundy domination, a vertex $v$ can be included in $S$ if it has a neighbor $u$ such that $u$ appears in the closed neighborhood of fewer than $k$ vertices of $S$. In this paper, we determine the $k$-Grundy domination number for some families of graphs, find degree-based bounds fo…

Intersecting families, cross-intersecting families, and a proof of a conjecture of Feghali, Johnson and Thomas

A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For a positive integer $n$, let $[n] = \{1, \dots, n\}$ and $\mathcal{S}_n = \{A \subseteq [n] \colon 1 \in A\}$. In this note, we extend the Erdős-Ko-Rado Theorem by showing that if $\mathcal{A}$ and $\mathcal{B}$ are non-empty cross-intersecting families of subsets of $[n]$, $\mathcal{A}$ is intersecting, and $a_0, a_1, \dots, a_n, b_0, b_1, \dot…

独立性、诱导子图和支配在K_{1,r}-自由图中

Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

本文研究了K_{1,r}-自由图中诱导子图的最大阶数与支配数的关系，得出了关于图的独立数和支配数的界，并通过Ramsey理论扩展了框架，证明了对于K_{1,r}-自由图的某些性质。