Zahraa Mohsen
A four blocks cycle C(k1,k2,k3,k4) is an oriented cycle formed by the union of four internally disjoint directed paths of lengths k1,k2,k3 and k4 respectively. El Mniny proved that if D is a digraph having a spanning out-tree T with no subdivisions of C(k, 1, 1, 1), then the chromatic number of D is at most 8^{3}k. In this paper, we will improve this bound to 18k.
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Cycle prefix digraphs have been proposed as an efficient model of symmetric interconnection networks for parallel architecture. It has been discovered that the cycle prefix networks have many attractive communication properties. In this paper, we determine the automorphism group of the cycle prefix digraphs. We show that the automorphism group of a cycle prefix digraph is isomorphic to the symmetric group on its underlying alphabet. Our method can be applied to other classes of graphs built on alphabets including the hypercube, the Kautz graph,and the de Bruijn graph.
Using some combinatorial techniques, in this note, it is proved that if $α\geq 0.28866$, then any digraph on $n$ vertices with minimum outdegree at least $αn$ contains a directed cycle of length at most 4.
Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree sequences which force $k$ vertex disjoint cycles and give the full answer for $k\le 2$.
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any $C$ a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of $C$. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree $2$ and two vertices have in-degree $2$).
The cycle space of a graph corresponds to the kernel of an incidence matrix. We investigate an analogous subspace for digraphs. In the case of digraphs of graphs, where every edge is replaced by two oppositely directed arcs, we give a combinatorial description of a basis of such a space. We are motivated by a connection to the transition matrices of discrete-time quantum walks.
Let $D$ be a strong digraph on $n=2m+1\geq 5$ vertices. In this paper we show that if $D$ contains a cycle of length $n-1$, then $D$ has also a cycle which contains all vertices with in-degree and out-degree at least $m$ (unless some extremal cases).
Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orient…
A weighted digraph is a digraph such that every arc is assigned a nonnegative number, called the weight of the arc. The weighted outdegree of a vertex $v$ in a weighted digraph $D$ is the sum of the weights of the arcs with $v$ as their tail, and the weight of a directed cycle $C$ in $D$ is the sum of the weights of the arcs of $C$. In this note we prove that if every vertex of a weighted digraph $D$ with order $n$ has weighted outdegree at least 1, then there exists a directed cycle in $D$ with weight at least $1/\log_2 n$. This proves a conjecture of Bollobás and Scott up to a constant facto…
A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles …
Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best…
The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order.
In 1985, Mader conjectured that for every acyclic digraph $F$ there exists $K=K(F)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of $F$. This conjecture remains widely open, even for digraphs $F$ on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader's problem and made the following conjecture: for every $\ell \geq 2$ there exists $K = K(\ell)$ such that every digraph $D$ with minimum out-degree at least $K$ contains a subdivision of every orientation of a cycle of length $\ell$. We prove this con…
Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.
Oriented Trees in Digraphs without Oriented $4$-cycles
本文证明了在最小半度至少为k/2且不含4-圈的有向图中，只要最大出度和入度至少为k，就能包含任意具有k条弧的有向树，并对反有向树和树状结构给出更强结论。
Sufficient conditions for a digraph to contain: a pre-Hamiltonian cycle and cycles of lengths 3 and 4
本文研究了有向图包含长度为3和4的回路以及前置哈密顿回路的充分条件，基于图的最小度数和半度数进行分析。
Subdivisions of Oriented Cycles in Digraphs with Hamiltonian directed path
针对有向图含指定有向圈细分的色数猜想，本文改进了哈密顿有向图情形的色数上界至$2n$，并将结果拓展到含哈密顿有向路径的有向图，深化了对猜想条件的认识。
Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sárközy and Selkow.
We prove a sharp Ore-type criterion for hamiltonicity of balanced bipartite digraphs: A bipartite digraph D, with colour classes of cardinality N, is hamiltonian if, for every pair of vertices u and v from opposite colour classes of D such that the arc uv is not in D, the sum of the outdegree of u and the indegree of v is greater than or equal to N+2.
We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in ${\cal D}(n,p)$ for nearly optimal $p$ (up to a $\log ^cn$ factor). In particular, we show that given $t = (1-o(1))np$ Hamilton cycles $C_1,\ldots ,C_{t}$, each of which is oriented arbitrarily, a digraph $D \sim {\cal D}(n,p)$ w.h.p. contains edge disjoint copies of $C_1,\ldots ,C_t$, provided $p=ω(\log ^3 n/n)$. We also show that given an arbitrarily oriented $n$-vertex cycle $C$, a random digraph $D \sim {\cal D}(n,p)$ w.h.p. contains $(1\pm o(1))n!p^n$ copies of $C$, provided $p \geq \log ^{1 + o(1)}n/n$.
Note on Long Directed Cycles in Eulerian Digraphs
本研究针对欧拉有向图中长有向环的长度下界问题，改进了Huang等人此前得出的下界结果，将其从$\sqrt{d}$提升至$\sqrt{2d}-3/2.$。
A minimum feedback arc set of a directed graph $G$ is a smallest set of arcs whose removal makes $G$ acyclic. Its cardinality is denoted by $β(G)$. We show that an Eulerian digraph with $n$ vertices and $m$ arcs has $β(G) \ge m^2/2n^2+m/2n$, and this bound is optimal for infinitely many $m, n$. Using this result we prove that an Eulerian digraph contains a cycle of length at most $6n^2/m$, and has an Eulerian subgraph with minimum degree at least $m^2/24n^3$. Both estimates are tight up to a constant factor. Finally, motivated by a conjecture of Bollobás and Scott, we also show how to find lon…
The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but …
We prove the following conjecture of S. Thomassé: for every (potentially infinite) digraph $ D $ it is possible to iteratively reverse directed cycles in such a way that the dichromatic number of the final reorientation $ D^{*} $ of $ D $ is at most two and each edge is flipped only finitely many times. In addition, we guarantee that in every strong component of $ D^{*} $ all the local edge-connectivities are finite and any edge is reversed at most twice.
Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.
Arc-disjoint Steiner Cycles in Digraphs
研究有向图中最大弧不相交S-Steiner环的数量，确定其在欧拉有向图、平面有向图和对称有向图上的计算复杂度，并给出完全有向图等特殊图类的精确值。
In the theory of digraphs, the study of cycles is a subject of great importance and has given birth to a number of deep questions such as the Behzad-Chartrand-Wall conjecture (1970) and its generalization, the Caccetta-Häggkvist conjecture (1978). Despite a lot of interest and efforts, the progress on these remains slow and mostly restricted to the solution of some special cases. In this note, we prove these conjectures for digraphs with girth is at least as large as their minimum out-degree and without short even cycles. More generally, we prove that if a digraph has sufficiently large girth …
Cycle tilings and $H$-factors in directed graphs
该研究提出了有向图中循环覆盖和H-因子的最小度数条件，并证明了关于传递竞赛因子的渐近精确Ore型结果。
When all directed cycles have the same weight
本文研究了所有有向环具有相同权重的有向图结构，提出两种构造方法并设计多项式时间算法来判断有向图是否可加权。
In this article, we evaluate determinants of block hook matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.
In 1992, Manoussakis conjectured that a strongly 2-connected digraph $D$ on $n$ vertices is hamiltonian if for every two distinct pairs of independent vertices $x,y$ and $w,z$ we have $d(x)+d(y)+d(w)+d(z)\geq 4n-3$. In this note we show that $D$ has a Hamilton path, which gives an affirmative evidence supporting this conjecture.
Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is $n/2+1$. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.
Reflexive Digraph Reconfiguration by Orientation Strings
针对环到自反有向图环的同态重构，论文用定向串关系刻画重构图连通分量，将复杂度由贪心算法的 Ω(m²) 改进为线性时间、对数空间。
Zero-Sum Cycles in Regular Digraphs
本文研究正则有向图中的零和环，证明了零和环存在的阈值，给出了顶点不交与边不交零和环的数量下界，结果渐近最优，相关结论可推广至欧拉有向图，所用技术基于Friedland的行列式-永久式论证。
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.
Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980's. It was finally shown by Frieze in 1986 that the random graph $\cG(n,p)$ with $p=c/n$ has a cycle on at all but at most $(1+ε)ce^{-c}n$ vertices with high probability, where $ε=ε(c)\to 0$ as $c\to\infty$. This estimate on the number of uncovered vertices is essentially tight due to vertices of degree 1. However, for the random digraph $\cD(n,p)$ no tight result was known and the best estimate was a factor of $c/2$ away from the corresponding lower bound. In this work we close this gap and show tha…
In 1972, Woodall raised the following Ore type condition for directed Hamilton cycles in digraphs: Let $D$ be a digraph. If for every vertex pair $u$ and $v$, where there is no arc from $u$ to $v$, we have $d^+u)+d^-(v)\geq |D|$, then $D$ has a directed Hamilton cycle. By a correspondence between bipartite graphs and digraphs, the above result is equivalent to the following result of Las Vergnas: Let $G = (B,W)$ be a balanced bipartite graph. If for any $b \in B$ and $w \in W$, where $b$ and $w$ are nonadjacent, we have $d(w)+d(b) \geq |G|/2 + 1$, then every perfect matching of $G$ is containe…
A path with three blocks $P(k,l,r)$ is an oriented path formed by $k$-forward arcs followed by $l$-backward arcs then $r$-forward arcs. We prove that any $(2k+1)$-chromatic digraph contains a path $P(1,k,1)$. However the existence of $P(1,l,1)$ with $l \geq k$ is established in any $(k+4)$-chromatic digraph. In general, we establish a quadratic bound for paths with three blocks.
In this paper we extend the concept of energy to signed digraphs. We obtain Coulson's integral formula for energy of signed digraphs. Formulae for energies of signed directed cycles are computed and it is shown that energy of non cycle balanced signed directed cycles increases monotonically with respect to number of vertices. Characterization of signed digraphs having energy equal to zero is given. We extend the concept of non complete extended $p$ sum (or briefly, NEPS) to signed digraphs. An infinite family of equienergetic signed digraphs is constructed. Moreover, we extend McClelland's ine…
The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemerédi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to suitable sparse robustly expanding digrap…
Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 10$ other than a directed cycle. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: If $ max\{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then $D$ contains cycles of all lengths $2,4, \ldots , 2a-2$ or $D$ is isomorphic to a certain digraph of order ten which we specify.
C. Thomassen (Proc. London Math. Soc. (3) 42 (1981), 231-251) gave a characterization of strongly connected non-Hamiltonian digraphs of order $p\geq 3$ with minimum degree $p-1$. In this paper we give an analogous characterization of strongly connected non-Hamiltonian digraphs with Meyniel-type condition (the sum of degrees of every pair of non-adjacent vertices $x$ and $y$ at least $2p-2$). Moreover, we prove that such digraphs $D$ contain cycles of all lengths $k$, for $2\leq k\leq m$, where $m$ is the length of a longest cycle in $D$.
Edge disjoint Hamilton cycles in random digraphs of constant minimum degree
研究随机图中在给定最小入度和出度条件下存在方向哈密顿回路的问题，证明在特定条件下图中存在k个边不相交的哈密顿回路。
Motivated by his work on the classification of countable homogeneous oriented graphs, Cherlin asked about the typical structure of oriented graphs (i) without a transitive triangle, or (ii) without an oriented triangle. We give an answer to these questions (which is not quite the predicted one). Our approach is based on the recent `hypergraph containers' method, developed independently by Saxton and Thomason as well as by Balogh, Morris and Samotij. Moreover, our results generalise to forbidden transitive tournaments and forbidden oriented cycles of any order, and also apply to digraphs. Along…
Orientations of cycles in digraphs of high chromatic number and high minimum out-degree
本文刻画了在色数足够大且最小出度与顶点数成比例的有向图中，哪些圈的定向必然作为子有向图出现，推广了 Thomassen 的已知结果。
Universality for rainbow oriented cycles in perturbed digraphs
研究随机扰动有向图在均匀n边着色下，高概率同时包含所有长度和定向的彩虹有向圈，通过Montgomery分布吸收法证明。
Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, Springer-Verlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979-992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284-302] that …
On oriented cycles in randomly perturbed digraphs
该研究推广了Bohman等人关于随机扰动有向图中哈密顿圈的结果，证明在最小半度条件下，渐近几乎必然同时存在所有长度和所有定向的圈，并可将条件放宽至最小总度情形，方法基于Montgomery吸收方法的变体。
The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in the underlying undirected graph of D such such that B,C are disjoint. It is known that there is a polynomial time algorithm for this problem when restricted to strongly connected graphs, which actually finds B,C if they exist. We generalize this to any class of digraphs D with either t(D) not equal to 1 or t(D)=1 and a bounded number o…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that addin…
We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph $K_{n[m]}^\ast$ with $n$ parts of size $m$ to admit a resolvable decomposition into directed cycles of length $t$. We show that the obvious necessary conditions are sufficient for $m,n,t \ge 2$ in each of the following four cases: (i) $m(n-1)$ is even; (ii) $\gcd(m,n) \not\in \{1,3\}$; (iii) $\gcd(m,n)=1$ and $4|n$ or $6|n$; and (iv) $\gcd(m,n)=3$, and if $n=6$, then $p|m$ for a prime $p \le 37$.
Let $k$ and $r$ be two integers with $k \ge 2$ and $k\ge r \ge 1$. In this paper we show that (1) if a strongly connected digraph $D$ contains no directed cycle of length $1$ modulo $k$, then $D$ is $k$-colorable; and (2) if a digraph $D$ contains no directed cycle of length $r$ modulo $k$, then $D$ can be vertex-colored with $k$ colors so that each color class induces an acyclic subdigraph in $D$. The first result gives an affirmative answer to a question posed by Tuza in 1992, and the second implies the following strong form of a conjecture of Diwan, Kenkre and Vishwanathan: If an undirected…
Covering Random Digraphs with Hamilton Cycles
本文研究了随机有向图的哈密顿圈覆盖问题，证明了在特定概率参数$p$范围内，随机有向图$D_{n,p}$可由最优数量的哈密顿圈覆盖，且该界限及对$p$的假设在多重对数因子内均达到最优。
For a simple digraph $G$ without directed triangles or digons, let $β(G)$ be the size of the smallest subset $X \subseteq E(G)$ such that $G\setminus X$ has no directed cycles, and let $γ(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. In 2008, Chudnovsky, Seymour, and Sullivan showed that $β(G) \le γ(G)$, and conjectured that $β(G) \le γ(G)/2$. Recently, Dunkum, Hamburger, and Pór proved that $β(G) \le 0.88 γ(G)$. In this note, we prove that $β(G) \le 0.8616 γ(G)$.
Let $D$ be a strong digraph on $n\geq 4$ vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) $d(x)+d(y)\geq 2n-1$ and $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this note we show that: if $D$ is not directed cycle and satisfies the condition (*), then $D$ contains a cycle of length $n-1$ or $n-2$.
Hamilton cycles in generalized dihedral Cayley graphs and digraphs
本文证明了广义二面体群上连通Cayley有向图存在有向Hamilton圈，证实了1978年的猜想，关键新工具是三次Haar图的三重和集覆盖定理。
A Note on Hamiltonian Cycles in Digraphs with Large Degrees
本文证明了阶为 $n$ 的 2-强有向图在满足特定顶点度数下界及存在经过特定顶点的长圈条件时必为哈密顿图。
Cycles of lengths 3 and n-1 in digraphs under a Bang-Jensen-Gutin-Li type conditon
本文研究有向图哈密顿性的Bang-Jensen-Gutin-Li型条件，建立了一个不仅蕴含哈密顿圈还蕴含3-圈和(n-1)-圈的条件，并猜想其蕴含所有长度圈的存在。
Let $D$ be a strong digraph on $n\geq 4$ vertices. In [2, J. Graph Theory 22 (2) (1996) 181-187)], J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (*) $d(x)+d(y)\geq 2n-1$ and $min \{d(x), d(y)\}\geq n-1$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or (**) $min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n$ for every pair of non-adjacent vertices $x, y$ with a common in-neighbour or a common out-neighbour, then $D$ is hamiltonian. In this paper we show that: (i) if $D$ satisfies the condition (*) and the minimum semi-degree of $D$ at least two or (i…
Transversal Hamilton cycles in digraph collections
本文研究了有向图集合中横贯哈密顿回路的存在性，推广了Ghouila-Houri定理，并应用了吸收方法、正则方法和横贯爆破引理等工具。
We discuss the length $\vec{L}_{c,n}$ of the longest directed cycle in the sparse random digraph $D_{n,p},p=c/n$, $c$ constant. We show that for large $c$ there exists a function $\vec{f}(c)$ such that $\vec{L}_{c,n}/n\to \vec{f}(c)$ a.s. The function $\vec{f}(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$ where $p_k$ is a polynomial in $c$. We are only able to explicitly give the values $p_1,p_2$, although we could in principle compute any $p_k$.
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