In this paper we investigate results of the form "every graph $G$ has a cycle $C$ such that the induced subgraph of $G$ on $V(G)\setminus V(C)$ has small maximum degree." Such results haven't been studied before, but are motivated by the Bessy and Thomassé Theorem which states that the vertices of any graph $G$ can be covered by a cycle $C_1$ in $G$ and disjoint cycle $C_2$ in the complement of $G$.
There are two main theorems in this paper. The first is that every graph has a cycle with $Δ(G[V(G)\setminus V(C)])\leq \frac12(|V(G)\setminus V(C)|-1)$. The bound on the maximum degree $Δ(G[V(G)\setminus V(C)])$ is best possible. The second theorem is that every $k$-connected graph $G$ has a cycle with $Δ(G[V(G)\setminus V(C)])\leq \frac1{k+1}|V(G)\setminus V(C)|+3$. We also give an application of this second theorem to a conjecture about partitioning edge-coloured complete graphs into monochromatic cycles.
We show that for any colouring of the edges of the complete bipartite graph $K_{n,n}$ with 3 colours there are 5 disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.
We show that for every η > 0 there exists an integer n_0 such that every 2-colouring of the 3-uniform complete hypergraph on n \geq n_0 vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover all but at most ηn vertices. The same result holds if we replace tight cycles with loose cycles.
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.
A classical result of Erdős, Gyárfás and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\ell$-cycles in different colours that together cover all but at most $4(k-\ell)$ vertices. If $\ell \leq k/3$, then at most two $\ell$-cycles cover all but at most $2(k-\ell)$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\ell\leq k/3$) disjoint monochromatic $\ell$-cycles.
For a graph $G$, let $σ_{2}(G)$ be the minimum degree sum of two non-adjacent vertices in $G$. A chord of a cycle in a graph $G$ is an edge of $G$ joining two non-consecutive vertices of the cycle. In this paper, we prove the following result, which is an extension of a result of Brandt et al. (J. Graph Theory 24 (1997) 165-173) for large graphs: For positive integers $k$ and $c$, there exists an integer $f(k,c)$ such that, if $G$ is a graph of order $n \ge f(k, c)$ and $σ_{2}(G) \ge n$, then $G$ can be partitioned into $k$ vertex-disjoint cycles, each of which has at least $c$ chords.
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further …
We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2α(G)$ monochromatic cycles, where $α(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1…
A conjecture of Erdős, Gyárfás, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gyárfás. As an intermediate result we s…
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and…
We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that i…
A cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\{v_i,v_{i+1}\}\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $λK_n^h$ be the $λ$-fold $n$-vertex complete $h$-graph. Let $\mathcal G=(V,E)$ be a hypergraph all of whose edges are of size at least $h$, and $2\leq c_1\leq \dots\leq c_k\leq |V|$. In order to partition the edge set of $\mathcal G$ into cycles of specified lengths $c_1, \dots, c_k$, an obvious necessary condition is that $\sum_{i=1}^k c_i=|E|$. We show that this condition is sufficient…
We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded.…
An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition.
An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erdős' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every …
We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random exponential time of parameter $q>0$. The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. This random partitioning is induced by a measure of rooted spanning forest of the graph, which generalizes the classical uniform spanning tree measure and which can be obtained as a zero-limit of FK-percolation with an external cemetery state. Some general properties of this rooted forest measure and related determin…
We prove that there exists a positive constant εsuch that if \log n / n \le p \le n^{-1+ε}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor δ(G)/2 \rfloor edge-disjoint Hamilton cycles.
We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph $G$ admits an $({\cal I}, {\cal O}_k)$-partition if its vertex set can be partitioned into an independent set and a set that induces a graph with components of order at most $k$. We prove that every graph $G$ with $\operatorname{mad}(G)<\frac 52$ admits an $({\cal I}, {\cal O}_3)$-partition. This implies that every planar graph with girth at least $10$ can be par…
Let $L_{c,n}$ denote the size of the longest cycle in $G(n,{c}/{n})$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $ L_{c,n}/n \to f(c)$ a.s. for $c\geq 20$, thus extending a result of the author and Frieze to smaller values of $c$. Thereafter, for $c\geq 20$, we determine the limit of the probability that $G(n,c/n)$ contains cycles of every length between the length of its shortest and its longest cycles as $n\to \infty$.
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…
In an earlier paper the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositi…
We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph $G=\gc$. In this model $G$ is drawn uniformly from graphs with vertex set $[n]$, $m$ edges and minimum degree at least three. We focus on the case where $m=cn$ for constant $c$. If $c$ is sufficiently large then our algorithm runs in $O(n^{1+o(1)})$ time and succeeds w.h.p.
Let $k \geq 2$ be an integer. Kouider and Lonc proved that the vertex set of every graph $G$ with $n \geq n_0(k)$ vertices and minimum degree at least $n/k$ can be covered by $k - 1$ cycles. Our main result states that for every $α> 0$ and $p = p(n) \in (0, 1]$, the same conclusion holds for graphs $G$ with minimum degree $(1/k + α)np$ that are sparse in the sense that \[ e_G(X,Y) \leq p|X||Y| + o(np\sqrt{|X||Y|}/\log^3 n) \qquad \forall X,Y\subseteq V(G). \] In particular, this allows us to determine the local resilience of random and pseudorandom graphs with respect to having a vertex cover …
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $G_{n,p}$ for $p \geq C(\log n/n)^{1/Δ}$, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph $H$ with $Δ(H) \leq Δ$. However, this is typically only optimal when $Δ\in \{2,3\}$ and $H$ either contains a triangle ($Δ= 2$) or many copi…
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 $p_k$.
We discuss the length of the longest cycle in a sparse random graph $G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function $f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$ where $p_k$ is a polynomial in $k$. We are only able to explicitly give the values $p_1,p_2$, although we could in principle compute any $p_k$. We see immediately that the length of the longest path is also asymptotic to $f(c)n$ w.h.p.
The {\em independence ratio} of a graph $G$ is defined by \[ ι(G) := \sup_{X \subset V(G)} \frac{|X|}{α(X)},\] where $α(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a relaxation of the chromatic number $χ(G)$ in the sense that $χ(G) \geq ι(G)$ for every graph $G$, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erdős on cycles in graphs with large chromatic number and a conjecture of Erdős and Hajnal on graphs with infinite chromatic number.
Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+ω(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in…
Let $L$ be subset of $\{3,4,\dots\}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1/2$ and the critical regime $M=\frac{n}{2}\left(1+μn^{-1/3}\right)$ with $μ=O(1)$. Depending on the regime and a condition involving the series $\sum_{l \in L} \frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to\infty$.
A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$ that is highly symmetric in the sense that every cycle can be derived from every other cycle just by permuting the axes. We conjecture that a similar decomposition exists for every 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…
We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $Γ$ be an $(n,d,λ)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-λ,λ]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $λ^{2k-1}\ll d^{2k}/n$, every subgraph $G$ of $Γ$ with $(1/2+o(1))e(Γ)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, Hàn, and the third author proved a weaker statement by allowing an extra polylogarithmic factor in the assumption $λ^{2k-1}\ll d^{2k}/n$, but we completely remove it and hence …
In 1962, Pósa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $δ(G)\ge 2n/3$. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $ε> 0$ and $p=n^{-1/2+ε}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+ε)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2…
As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $Δ(H)$, then $H$ has a vertex $Δ(H)$-coloring such that one of the color classes is a maximum independent set. Let $G$ be a connected graph of order at least $2$. A $G$-free $k$-coloring of a graph $H$ is a partition of the vertex set of $H$ into $V_1,\ldots,V_k$ such that $H[V_i]$, the subgraph induced on $V_i$, does not contain any subgraph isomorphic to $G$. As a generalization of Catlin's theorem we show that a graph $H$ has a $G$-free $\lceil{Δ(H)\over …
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of deter…
Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）；ECOPRO, Institute for Basic Science（基础科学研究院 ECOPRO）
We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Γ$ the generalized Turán density $π_F(Γ)$ denotes the density of a maximum subgraph of $Γ$, which contains no copy of~$F$. Extending classical Turán type results for odd cycles, we show that $π_{F}(Γ)=1/2$ provided $F$ is an odd cycle and $Γ$ is a sufficiently pseudorandom graph. In particular, for $(n,d,λ)$-graphs $Γ$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-λ,λ]$, our result holds for odd cycles of length $\ell$, provided \[ λ^{\ell-2}\ll \frac{d^{\ell-1}}n…
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as the (full) correlation polytope) $\mathscr{P}(G)$ of the complete graph $G=K_n$. We extend this work to structured sparse graphs, giving: (i) an efficient algorithm for $vol(\mathscr{Q}(G))$ when $G$ has bounded tree width, (ii) closed-form expressions (and asymptotic behaviors) for $vol(\mathscr{Q}(G))$ for all stars, paths, and cycles, and (iii) a closed-…
We prove that the number of Hamilton cycles in the random graph G(n,p) is n!p^n(1+o(1))^n a.a.s., provided that p\geq (ln n+ln ln n+ω(1))/n. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e)^n(1+o(1))^n Hamilton cycles a.a.s.
Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =Ω(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length Ω(d^{\lfloor (g-1)/2\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be im…
Supersaturation of induced even cycles in locally sparse graphs
Charles University, Faculty of Mathematics and Physics（查理大学数学物理学院）；EPFL（洛桑联邦理工学院）；University of Oxford（牛津大学）；École Polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎理工学院）
Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4-cycle in the binomial random graph $G_{n,p}$. We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain (large) range of $p$, the upper tail of the induced 4-cycle does not admit a naive mean-field approximation.
We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big…
This paper presents the results of an experimental study of graph partitioning. We describe a new heuristic technique, path optimization, and its application to two variations of graph partitioning: the max_cut problem and the min_quotient_cut problem. We present the results of computational comparisons between this technique and the Kernighan-Lin algorithm, the simulated annealing algorithm, the FLOW-lagorithm the multilevel algorithm, and teh recent 0.878-approximation algorithm. The experiments were conducted on two classes of graphs that have become standard for such tests: random and rand…
A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of $H$-free graphs, that is, graphs that do not contain some graph $H$ as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph $H$ contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study on the $k$-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the $H$-free input graph. We continue this line of research by …
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem …
Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides…
We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erdős--Rényi random graphs on $n$ vertices with a fixed average degree $c>1$. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in $n$. Here we derive, for a random $d$-regular graph, the limiting probability that ${\cal L}(G)$ simultaneously contains the entire range $\{\ell,\ldots,n\}$ fo…
