Motivated by Thomassen's well-known line graph conjecture, many researchers have explored sufficient conditions for claw-free graphs to be Hamiltonian or Hamilton-connected. In 1994, Ageev proved that every $2$-connected claw-free graph with domination number at most $2$ is Hamiltonian. In this paper, we extend this line of research to $3$-connected graphs by establishing the best possible upper bound on the domination number that guarantees Hamiltonicity. Specifically, we show that, except for some well-defined exceptional graphs, every $3$-connected claw-free graph $G$ with domination number at most $5$ is Hamiltonian. Furthermore, we prove that, apart from a few exceptional cases, every $3$-connected claw-free graph $G$ with domination number at most $4$ is Hamilton-connected, thereby generalizing earlier results of Zheng, Broersma, Wang and Zhang and Vrána, Zhan and Zhang. We further investigate the Hamiltonian properties of line graphs of $3$-hypergraphs, and prove that every 3-connected line graph of a 3-hypergraph with domination number at most $4$ is Hamiltonian.
In 2006 Bar{á}t and Thomassen conjectured that every planar $4$-edge-connected $4$-regular simple graph of size divisible by three admits a claw-decomposition. Later, Lai (2007) disproved this conjecture by a family of planar graphs with edge-connectivity $4$ which the smallest one contains $24$ vertices. In this note, we first give a smaller counterexample having only $18$ vertices and next construct a family of planar $4$-connected essentially $6$-edge-connected $4$-regular simple graphs of size divisible by three with no claw-decompositions. This result provides the sharpness for two known …
Coloring clique-hypergraph of $K_5$-minor-free graphs
图$G$的团染色是对图$G$的顶点的染色，使得大小至少为2的最大团都不是单色的。图$G$的群超图$\mathcal{H}(G)$以$V(G)$为其顶点集，以$G$的最大团为其超边。$\mathcal{H}(G)$的一个（顶点）染色是$G$的一个顶点染色。$G$的团色数是$G$允许团色的最少颜色数。任何可平面图都是3-可色的（Electr. J. Combin. 6（1999），\#R26）。最近，我们证明了每个不同于奇圈的无爪可平面图都是$2$ -可染的（European J. Combin. 36（2014）367-376）。本文将这些结果推广到无爪，无$K_5$ -子图.
st-路径是一条端点为s和t的路径。s-路径是一条端点为s的路径。本文的结果包括给定两个不同顶点s，t和一条边e的无{爪，网}图G有（1）Hamilton s-路，（2）Hamilton st-路，（3）连通度为1时，有含边e的Hamilton s-路和st-路，（4）2-连通时，有含边e的哈密顿圈的充要条件.这些结果意味着一个连通的无{claw，net}图有一条Hamilton路，一个2连通的无{claw，net}图有一个哈密顿圈[D. Duffus，R. J. Gould，M. S. Jacobson，Forbidden Subgraphs and the Hamilton Theme，in The Theory and Application of Graphs（Kalamazoo，Mich.，1980 $），Wiley，New York（1981）297 - 316.]我们对（1）-（4）的证明比[D. Duffus，R. J. Gould，M. S. Jacobson]，并给出了求解相应Hamilton性问题的多项式时间算法。关键词：图，爪，网，无{爪，网}图，哈密尔顿路，哈密尔顿圈，多项式时间算法。
本文首先证明了连通$4$ -正则无爪图在$n$顶点上的幂控制数至多为$\frac{n+1}{5}$，且有界是尖锐的.该声明部分反驳了Dorbec等人在SIAM J. Discrete Math.中提出的猜想，27：1559-1574，2013.在此基础上，我们提出了一个求解树的加权幂控制问题的动态规划式线性时间算法。
图G的一个L-因子是G的一个生成子图，它的每个分支都是一条3-顶点路，设v（G）表示G的顶点数。一个图称为无爪图，如果它没有一个子图同构于具有4个顶点和3条边有一个公共顶点的图。我们的研究结果包括以下内容。设G$是一个3-连通无爪图，x是G中的一个点，e = xy是G中的一条边，P是G中的一条3-点路。则（c1）如果v（G）= 0 mod 3，则G有一个L-因子，（避免）e，（c2）如果v（G）= 1 mod 3，则G-x具有L-因子，（c3）如果v（G）= 2 mod 3，则G-x-y具有L-因子，（c4）如果v（G）= 0 mod 3并且G是立方的或4-连通的，则G-P具有L-因子，（c5）若G是三次图，且E是G中的一个三边集，则G-E有L -因子当且仅当E在G中导出的子图不是爪形图也不是三角形图.关键词：无爪图，三次图，L-填充，L-因子。
In this paper, we give a series of couterexamples to negate a conjecture and hence answer an open question on the $k$-power domination of regular graphs (see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]). Furthermore, we focus on the study of $k$-power domination of claw-free graphs. We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most $\frac{n}{k+l+2}$, and this bound is tight.
In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are $2$-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are $2$-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (…
We generalize a result of Balister, Gy{ő}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a $k$-Berge-path, where $n \geq N_{k,r}$, $k\geq 2r+13>17$.
We prove that every 3-connected claw-free graph with domination number at most 3 is hamiltonian-connected. The result is sharp and it is inspired by a conjecture posed by Zheng, Broersma, Wang and Zhang in 2020.
An $r$-uniform hypergraphic sequence (i.e., $r$-graphic sequence) $d=(d_1, d_2,\cdots,d_n)$ is said to be forcibly $k$-edge-connected if every realization of $d$ is $k$-edge-connected. In this paper, we obtain a strongest sufficient degree condition for $d$ to be $k$-edge-connected for all $k\ge 1$ and a strongest sufficient degree condition for $d$ to be super edge-connected. As a corollary, we give the minimum degree condition for $d$ to be maximally edge-connected. We also obtain another sufficient degree condition for $d$ to be $k$-edge-connected.
A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have $K_{1,c}$ as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to th…
For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$ is an induced subgraph of some of $P_6$, $K_1 \cup P_5$, $2P_3$, $Z_2$ or $K_1 \cup Z_1$. Furthermore, for $X$ chosen as $2K_1 \cup K_3$, we list all eight imperfect graphs belonging to the class; and for every other choice of $X$, we show that there are infinitely many such graphs. In addition, for $X$ chosen as $B_{1,2}$, we describe the structure of all …
