Alexander Kelmans
We give a construction that provides infinitely many 2-connected, cubic, bipartite, and planar graphs G with 3k vertices and such that the number of disjoint copies of a 3-vertex path in G is less than k.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Given a graph $G$, a function $c:V(G)\longrightarrow \{1,\ldots,k\}$ with the property that $c(u)=c(v)=i$ implies that the distance between $u$ and $v$ is greater than $i$, is called a $k$-packing coloring of $G$. The smallest integer $k$ for which there exists a $k$-packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ$. Packing chromatic vertex-critical graphs are the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. A graph $G$ is called a packing chromatic critical graph if for every proper subgraph $H$ of $G$, $χ_ρ(H) < χ_ρ…
Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least v(G)/(s+1), (a2) if G is a graph with every vertex of degree at least two and at most 3 and G has no 5-vertex components, then t(G,2) is at least v(G)/4, (a3) if G is a graph with every vertex of degree at least one and at most s and G has no k--vertex component, where k >1 and s > 2, then t(G,k) is at least (v(G) - k)/(sk - k +1), and (a4) the above bounds are…
We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a sufficient Hamiltonicity condition for the line graphs of such graphs.
Let v(G) and p(G) be the number of vertices and the maximum number of disjoint 3-vertex paths in G, respectively. We discuss the following old Problem: Is the following claim (P) true ? (P) if G is a 3-connected and cubic graph, then p(G) = [v(G)/3], where [v(G)/3] is the floor of v(G)/3. We show, in particular, that claim (P) is equivalent to some seemingly stronger claims. It follows that if claim (P) is true, then Reed's dominating graph conjecture (see [14]) is true for cubic 3-connected graphs.
The packing number of the double vertex graph of the path graph
本文求出路径图双顶点图（即 Sloane 图）的装填数，解决了可纠正单个相邻换位的恒权2二进制码最大规模这一公开问题，并证实 Rob Pratt 关于数列 A085680 生成函数的猜想。
A subgraph (a spanning subgraph) of a graph G whose all components are 3-vertex paths is called an L-packing (respectively, an L-factor} of G. We discuss the following old PROBLEM (A. Kelmans, 1984). Is the following claim true? (C) If G is a cubic 3-connected graph, then G has an L-packing that avoids at most two vertices of G. We show, in particular, that claim (C) is equivalent to some seemingly stronger claims (see Theorem 3.1 below). For example, if G is a cubic 3-connected graph and the number of vertices of G is divisible by three, then then the following claims are equivalent: G has an…
A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $γ(G)$, is the minimum order of a dominating set. A subset $R$ of edges of a graph $G$ is a 2-degree-packing, if any three edges from $R$ do not have the same incident vertex. The 2-degree-packing number of $G$, $ν_2(G)$, is the maximum order of a 2-degree-packing of $G$. In this paper, we prove that any simple graph $G$ satisfies $γ(G)\leqν_2(G)-1$. Furthermore, we give a characterization of simple connected graphs $G$ sat…
Broadcast independence and packing in certain classes of trees
本文研究完美 k 叉树、蜘蛛图和双蜘蛛图中最大权重的独立广播与 packing 广播问题，部分回答了 Ahmane 等人提出的问题。
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices and 3 edges having a common vertex. Our results include the following. Let G$ be a 3-connected claw-free graph, x be a vertex, e = xy be an edge, and P be a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (c2) if v(G) = 1 mod 3, then G - x has a L-factor, (c3) if v(G) = 2 mod 3, then G - x -y has an L-factor, (…
Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)…
A characterization of 4-$χ_S$-vertex-critical graphs for packing sequences with $s_1 =1$ and $s_2\ge 3$
本文刻画了当打包序列满足$s_1=1$且$s_2\ge 3$时，$4$-色数顶点临界图的结构，确定了28个零星例子和两个无限族。
An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (a3…
Impact of local girth on the S-packing coloring of k-saturated subcubic graphs
研究局部圈长对k饱和子三次图S-打包着色能力的影响，通过参数分析和构造验证，揭示饱和度与局部圈长共同决定的可接受S-打包序列。
$S$-packing chromatic vertex-critical graphs
本文研究了 $S$-packing 色数顶点临界图的性质，刻画了 3-和 4-临界图，并探讨了树和毛毛虫图的临界性。
A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if every component of the spanning subgraph is isomorphic to the star graph $K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of $G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph is a connected plane cubic graph with only pentagonal and hexagonal faces, which is the molecular graph of carbon fullerene. Clearly, a perfect star packing in a fullerene graph $G$ on $n$ vertices will exist if and…
We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph.
Limited packings: related vertex partitions and duality issues
本文研究了图的最小 $k$-有限包装划分问题，给出了树中该参数的精确值与一般图的界限，解决了1998年提出的相关对偶开放问题，并证明计算总有限包装数是 NP-hard 的。
A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph $H$, there exists a function $f$ such that for every 4-edge-connected graph $G$, either $G$ contains $k$ pairwise edge-disjoint subgraphs each containing $H$ as an immersion, or there exists a set of at most $f(k)$ edges of $G$ intersecting all such subgraphs. …
Let $G=(V,E)$ and $H$ be two graphs. Packing problem is to find in $G$ the largest number of independent subgraphs each of which is isomorphic to $H$. Let $U\subset{V}$. If the graph $G-U$ has no subgraph isomorphic to $H$, $U$ is a cover of $G$. Covering problem is to find the smallest set $U$. The vertex-disjoint tree packing was not sufficiently discussed in literature but has its applications in data encryption and in communication networks such as multi-cast routing protocol design. In this paper, we give the kind of $(k+1)$-connected graph $G'$ into which we can pack independently the su…
We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where N[v] is the closed neighbourhood of v. The k-limited packing number $L_k(G)$ of a graph G is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks. In this paper, we develop a new probabilistic approach to limited packings in graphs, resulting in lower bounds for the k-limited packing number and a randomized algo…
Let $G$ be a graph and let $l$ be an integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. We say that $G$ is $l$-rigid, if it contains a spanning $l$-partition-connected subgraph $H$ with $|E(H)|=\sum_{v\in V(H)} l(v)-l(V(H))$. In this paper, we investigate decomposition of graphs into spanning partition-connected and spanning rigid subgraphs. As a consequence, we improve a recent result due…
Packing chromatic vertex-critical graphs
本文引入打包染色顶点临界图（$χ_ρ$-临界图），刻画了3-和4-临界图，证明任意$k\ge2$存在$k$-临界树，$k$-临界毛毛虫当且仅当$k\le7$，并给出笛卡尔积为临界图的充分条件。
We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and Rödl.
S-packing chromatic critical paths and cycles
本文确定了每条路径的S-打包着色数，并识别了对应的临界路径和顶点临界路径。同时，扩展了关于S-打包临界环的先前结果，给出了完整的特征化。
Simis and packing properties of Alexander dual of connected ideals
本文研究了连通图理想亚历山大对偶的普通幂与符号幂何时相等，并对这类图进行了完整分类，同时证明了Conforti-Cornuèjols猜想。
The \textit{packing chromatic number} of a graph $G$, denoted by $% χ_ρ(G)$, is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we present some general properties of packing chromatic numbers of \textit{finite super subdivisions} of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and \textit{neighborhood corona graphs} of a cycle and a path respectively of a complete graph and a path.
For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4 and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the following two properties: * for every graph G with V(G)=V(H), at least p proportion of the triangles in G are also edges of H, * for every vertex x of H, the link graph of x is a quasirandom graph with density at le…
Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $δ(H, n)$ be the minimum $δ$ such that any $k$-graph $G$ on $n$ vertices with minimum codegree $δ(G) \geq δ$ contains a perfect $H$-packing. The problem of determining $δ(H, n)$ has been widely studied for graphs (i.e. $2$-graphs), but little is known for $k \geq 3$. Here we determine the asymptotic value of $δ(H, n)$ for all complete $k$-partite $k$-graphs $H$, as well as a wide class …
We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jordán [4] and a theorem of Jordán [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lovász and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-con…
A Short Proof that Every Claw-Free Cubic Graph is (1,1,2,2)-Packing Colorable
本文给出了无爪立方图是(1,1,2,2)-打包着色的简短证明，改进了之前的结果。
On the Packing Coloring Gap of Graphs
本文研究了图的打包着色间隙，定义为删除单个顶点导致打包着色数最大减少的值，重点分析了树结构，特别是毛毛虫图，并扩展到毛毛虫图与K1的冠操作，同时给出了具有不同间隙值的图示例。
We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).
Every connected subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable
本文证明了所有连通次立方图除了彼得森图都是可(1,1,2,2)-着色的，解决了相关猜想和开放问题。
Graphic additive bases and graph flows
本文通过生成树打包研究 Tutte 流猜想，提出指定方向定理并证明图论加法基猜想，同时给出圆周流界及反例说明打包数大于 9/2 不足以保证模五方向。
Domination and packing in graphs
研究图类中支配数与覆盖数比值的上界，证明平面图的比值上界改进，并扩展了适用于弦 bipartite 图和同质可序图的结论。
Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoin…
Packing subdivisions into regular graphs
该研究证明了在足够大的正则图中，可以找到覆盖大部分顶点的子划分，验证了已有猜想并改进了相关结果。
We consider (closed neighbourhood) packings and their generalization in graphs called limited packings. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where $N[v]$ is the closed neighbourhood of $v$. The k-limited packing number $L_k(G)$ is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks. We develop probabilistic and greedy approaches to limited packings in graphs, providing lower bounds for the k-limited packing number, and randomized a…
Packing Directed Cycles Quarter- and Half-Integrally
本研究针对有向图的环装填与反馈顶点集问题，证明了四分之一积分和二分之一积分环装填数下反馈顶点集大小的多项式界，并得到高有向树宽子图四分之一积分装填的一般性结果。
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. In this short note we present upper and lower bound for the packing chromatic number of the lexicographic product $G\circ H$ of graphs $G$ and $H$. Both bounds coincide in many cases. In particular this happens if $|V(H)|-α(H)\geq {\rm diam}(G)-1$, where $α(G)$ denotes the independence number of $G$.
A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum deg…
An edge (vertex) cut $X$ of $G$ is $r$-essential if $G-X$ has two components each of which has at least $r$ edges. A graph $G$ is $r$-essentially $k$-edge-connected (resp. $k$-connected) if it has no $r$-essential edge (resp. vertex) cuts of size less than $k$. If $r=1$, we simply call it essential. Recently, Lai and Li proved that every $m$-edge-connected essentially $h$-edge-connected graph contains $k$ edge-disjoint spanning trees, where $k,m,h$ are positive integers such that $k+1\le m\le 2k-1$ and $h\ge \frac{m^2}{m-k}-2$. In this paper, we show that every $m$-edge-connected and $2$-essen…
For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $χ_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$…
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ a…
A linear upper bound on the $\mathbb{Z}_p$-Ramsey number of graphs with sufficiently large $2$-packing
本文研究了具有足够大2-打包的图的Z_p-拉姆齐数，证明了对于具有特定条件的图G，其Z_p-拉姆齐数R(G,Z_p)不超过n+6p-9，并进一步得出对于最大度有界的图，R(G,Z_p)的上界形式为R(G,Z_p)≤n+C。
We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect H-packings: Given positive integers n, r, D, we characterise the edge density threshold that ensures a perfect K_r-packing in any graph G on n vertices and with minimum degree at least D. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect H-packing. Other related embedding problems are also considered. Indeed, we give a structural result concerning K_r-free g…
In this paper, we investigate the packing parameters in graphs. By applying the Mantel's theorem, We give upper bounds on packing and open packing numbers of triangle-free graphs along with characterizing the graphs for which the equalities hold and exhibit sharp Nordhaus-Gaddum type inequalities for packing numbers. We also solve the open problem of characterizing all connected graphs with $ρ_{o}(G)=n-ω(G)$ posed in [S. Hamid and S. Saravanakumar, {\em Packing parameters in graphs}, Discuss Math. Graph Theory, 35 (2015), 5--16].
Covering and 2-degree-packing numbers in graphs
研究简单图覆盖数与新提出的2-度装填数之间的关系，证明了在边数条件下两者的不等式界，并刻画了取等图类。
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。
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. It is proved that in the class of subcubic graphs the packing chromatic number is bigger than $13$, thus answering an open problem from [Gastineau, Togni, $S$-packing colorings of cubic graphs, Discrete Math.\ 339 (2016) 2461--2470]. In addition, the packing chromatic number is investigated with respect to several local operations. In particular, if $S_e(G)$ is the graph obtained from a…
New lower bounds on domination--packing ratios in connected subcubic and cubic graphs
该研究针对连通次立方图和立方图，构造显式二元分支族，证明其支配数与填充数之比的下界分别为13/6和17/8，反驳了\textit{γ(G)≤2ρ(G)+1}对连通立方图的适用性。
In this paper, we find upper bounds on the open packing and $k$-limited packing numbers with emphasis on the cases $k=1$ and $k=2$. We solve the problem of characterizing all connected graphs on $n$ vertices with $ρ_{o}(G)=n/δ(G)$ which was presented in 2015 by Hamid and Saravanakumar. Also, by establishing a relation between the $k$-limited packing number and double domination number we improve two upper bounds given by Chellali and Haynes in 2005.
The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by $8$ in the family of Sierpiński graphs with base $3$, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than $3$. On the other hand, we prove that the packing chromatic number in the family of …
Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is often the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete. Indeed, if $k \geq 3$, the corresponding problem for perfect matchings is NP-complete whilst if $k=2$ the problem is NP-complete in the case when $F$ has a component consisting of at least $3…
Packing mixed hyperarborescences
本文提出了一个新的定向定理，简化证明了混合图中基于拟阵可达性的混合树打包结果，并扩展了关于混合超图中混合超树打包的新定理。
Let $G$ be a graph with $X\subseteq V(G)$ and let $l$ be an intersecting supermodular subadditive integer-valued function on subsets of $V(G)$. The graph $G$ is said to be $l$-partition-connected, if for every partition $P$ of $V(G)$, $e_G(P)\ge \sum_{A\in P} l(A)-l(V(G))$, where $e_G(P)$ denotes the number of edges of $G$ joining different parts of $P$. Let $λ\in [0,1]$ be a real number and let $η$ be a real function on $X$. In this paper, we show that if $G$ is $l$-partition-connected and for all $S\subseteq X$, $$Θ_l(G \setminus S) \le \sum_{v\in S} (η(v) -2l(v))+l(V(G))+l(S)-λ(e_G(S))+l(S)…
Packing and Covering Cycles Through Prescribed Vertices
该研究针对有限简单图，证明与指定顶点子集相交的环所需的最少顶点数不超过顶点不相交环覆盖该子集的最大顶点数，解答了相关学者的问题，通过归约方法完成论证。
Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs
研究了在渗流顶点转置图中，通过概率p保留边后，保证存在完美匹配和2-因子的条件及概率。
Approximate packing of independent transversals in locally sparse graphs
本文研究局部稀疏多部图中独立横截集的分解猜想，证明可找到 $(1-\varepsilon)n$ 个两两不交的独立横截集，近似解决该猜想并应用于相关问题。
On uniqueness of packing of three copies of 2-factors
本文完全解决了2-正则图三个副本打包的唯一性问题，给出了无打包、唯一打包和至少两种打包的完整分类。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。