Odd Cycle Of Length Research Articles

Odd Holes in Bull-Free Graphs

The complexity of testing whether a graph contains an induced odd cycle of length at least five is currently unknown. In this paper we show that this test can be done in polynomial time if the inpu...

On the character degree graph of solvable groups

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. Palfy asserts that the complement $\bar{\Delta}(G)$ of the graph \(\Delta(G)\) does not contain any cycle of length \(3\). In this paper we generalize Palfy's result, showing that $\bar{\Delta}(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of \(\Delta(G)\) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if \(n\) is the clique number of \(\Delta(G)\), then \(\Delta(G)\) has at most \(2n\) vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous \emph{\(\rho\)-\(\sigma\) conjecture} by B. Huppert.

Completing the spectrum of almost resolvable cycle systems with odd cycle length

In this paper, we construct almost resolvable cycle systems of order 4k+1 for odd k≥11. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton–Waterloo problem are also obtained.

A Note on Chromatic Number and Induced Odd Cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph $G$ has no odd holes then $\chi(G)\le 2^{2^{\omega(G)+2}}$. Chudnovsky, Robertson, Seymour and Thomas showed that if $G$ has neither $K_4$ nor odd holes then $\chi(G)\le 4$. In this note, we show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then $\chi(G)\le 4$ and $\chi(G)\le 3$ if $G$ has radius at most $3$, and for each vertex $u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions in Plummer and Zha (2014).

On the Hamilton–Waterloo problem with odd cycle lengths

AbstractLet denote the complete graph if v is odd and , the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers , the Hamilton–Waterloo problem asks for a 2‐factorization of into α ‐factors and β ‐factors, with a ‐factor of being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, , , and are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd the above necessary conditions are sufficient, except possibly when , or when . Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

Completely positive semidefinite rank

An $n\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\times d$ Hermitian positive semidefinite matrices $\{P_i\}_{i=1}^n$ (for some $d\ge 1$) such that $X_{ij}= {\rm Tr}(P_iP_j),$ for all $i,j \in \{ 1, \ldots, n \}$. The cpsd-rank of a cpsd matrix is the smallest $d\ge 1$ for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior. In this work we present several properties of the cpsd-rank. Unlike the completely positive rank which is at most quadratic in the size of the matrix, no general upper bound is known on the cpsd-rank of a cpsd matrix. In fact, we show that the cpsd-rank can be exponential in terms of the size. Specifically, for any $n\ge1,$ we construct a cpsd matrix of size $2n$ whose cpsd-rank is $2^{\Omega(\sqrt{n})}$. Our construction is based on Gram matrices of Lorentz cone vectors, which we show are cpsd. The proof relies crucially on the connection between the cpsd-rank and quantum behaviors. In particular, we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high-rank extreme points of the $n$-dimensional elliptope. Lastly, we study cpsd-graphs, i.e., graphs $G$ with the property that every doubly nonnegative matrix whose support is given by $G$ is cpsd. We show that a graph is cpsd if and only if it has no odd cycle of length at least $5$ as a subgraph. This coincides with the characterization of cp-graphs.

Further results on almost resolvable cycle systems and the Hamilton–Waterloo problem

AbstractIn this paper, we almost completely solve the existence of almost resolvable cycle systems with odd cycle length. We also use almost resolvable cycle systems as well as other combinatorial structures to give some new solutions to the Hamilton–Waterloo problem.

New characterizations of Gallai’s i-triangulated graphs

The i-triangulated graphs, introduced by Tibor Gallai in the early 1960s, have a number of characterizations—one of the simplest is that every odd cycle of length 5 or more has noncrossing chords. A variety of new characterizations are proved, starting with a simple forbidden induced subgraph characterization and including the following two:(1) Every 2-connected induced subgraph H is bipartite or chordal or, for every induced even cycle C of H, the intersection of the neighborhoods in H of all the vertices of C induces a complete multipartite subgraph.(2) For every 2-connected induced subgraph H that is not bipartite, every cardinality-2 minimal vertex separator of H induces an edge.

Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most d and how excluding one fixed cycle length ℓ affects the number of colours required as d→∞. For vertex-colouring and t≥1, if any two distinct vertices connected by a path of at most t edges are required to be coloured differently, then a reduction by a logarithmic (in d) factor against the trivial bound O(dt) can be obtained by excluding an odd cycle length ℓ≥3t if t is odd or by excluding an even cycle length ℓ≥2t+2. For edge-colouring and t≥2, if any two distinct edges connected by a path of fewer than t edges are required to be coloured differently, then excluding an even cycle length ℓ≥2t is sufficient for a logarithmic factor reduction. For t≥2, neither of the above statements are possible for other parity combinations of ℓ and t. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

Infinite kernel perfect digraphs

Let D be a digraph, possibly infinite, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A subset K of V(D) is said to be a kernel if it is both independent (a vertex in K has no successor in K) and absorbing (a vertex not in K has a successor in K). An infinite digraph D is said to be a finitely critical kernel imperfect digraph if D contains no kernel but every finite induced subdigraph of D contains a kernel. In this paper we will characterize the infinite kernel perfect digraphs by means of finitely critical imperfect digraphs and strong components of its asymmetric part and then, by using some previous theorems for infinite digraphs, we will deduce several results from the main result. Richardson’s theorem establishes that if D is a finite digraph without cycles of odd length, then D has a kernel. In this paper we will show a generalization of this theorem for infinite digraphs.

Ramsey Numbers of Trees Versus Odd Cycles

Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.

Infinitely many cyclic solutions to the Hamilton–Waterloo problem with odd length cycles

It is conjectured that for every pair (ℓ,m) of odd integers greater than 2 with m≡1(modℓ), there exists a cyclic two-factorization of Kℓm having exactly (m−1)/2 factors of type ℓm and all the others of type mℓ. The authors prove the conjecture in the affirmative when ℓ≡1(mod4) and m≥ℓ2−ℓ+1.

Induced subgraphs of graphs with large chromatic number. I. Odd holes

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no Kt subgraph and no odd hole is n-colourable. We prove this conjecture.

Coloring signed graphs using DFS

We show that depth first search can be used to give a proper coloring of connected signed graphs G using at most $$\Delta (G)$$ colors, provided G is different from a balanced complete graph, a balanced cycle of odd length, and an unbalanced cycle of even length, thus giving a new, short proof to the generalization of Brooks’ theorem to signed graphs, first proved by Macajova, Raspaud, and Skoviera.

Doyen–Wilson Results for Odd Length Cycle Systems

AbstractFor each odd , we completely solve the problem of when an m‐cycle system of order u can be embedded in an m‐cycle system of order v, barring a finite number of possible exceptions. In cases where u is large compared to m, where m is a prime power, or where , the problem is completely resolved. In other cases, the only possible exceptions occur when is small compared to m. This result is proved as a consequence of a more general result that gives necessary and sufficient conditions for the existence of an m‐cycle decomposition of a complete graph of order v with a hole of size u in the case where and both hold.

Formula omitted]-kernels and [formula omitted]-obstructions in [formula omitted]-colored digraphs

Let D be a digraph. V(D) and A(D) will denote the vertex and arc sets of D respectively. A kernel K of a digraph D is an independent set of vertices of D such that for every vertex w in V(D)-K there exists an arc from w to a vertex in K. Let H be a digraph possibly with loops and D a digraph without loops whose arcs are colored with the vertices of H (D is said to be an H-colored digraph). A directed path W in D is said to be an H-path if and only if the consecutive colors encountered on W form a directed walk in H. A generalization of the concept of kernel is the concept of H-kernel, where an H-kernel N of an H-colored digraph D is a set of vertices of D such that for every pair of different vertices in N there is no H-path between them, and for every vertex u in V(D)-N there exists an H-path in D from u to N. A classical result in kernel theory establishes that if D is a digraph without cycles of odd length, then D has a kernel; this result is known as Richardson’s theorem and in this paper we will show an extension of this theorem which is given by the main result.Let D be an H-colored digraph. For an arc (z1, z2) of D we will denote its color by c(z1, z2). We introduce the concept of obstruction in an H-colored digraph as follows. Let W=(v0,v1,…,vn−1,v0) be a closed directed walk in D. We will say that there is an obstruction on vi if (c(vi−1, vi), c(vi, vi+1)) is not an arc of A(H) (indices modulo n). The main result establishes that if D is an H-colored digraph such that the number of obstructions in every closed directed trail of D is even, then D has an H-kernel. Previous interesting results are generalized, as for example Sands, Sauer and Woodrow’s theorem.

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

AbstractA graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. (http://arxiv.org/abs/1308.6444).

Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant

The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. An n-dimensional folded hypercube (FQn for short) for any odd n is known to be bipartite. In this paper, let f be a faulty vertex in FQn. It has been shown that (1) Every edge of FQn−{f} lies on a fault-free cycle of every even length l with 4≤l≤2n−2 where n≥3; (2) Every edge of FQn−{f} lies on a fault-free cycle of every odd length l with n+1≤l≤2n−1, where n≥2 is even. In terms of every edge lies on a fault-free cycle of every odd length in FQn−{f}, our result improves the result of Cheng et al. (2013) where odd cycle length up to 2n−3.

Extremal results for odd cycles in sparse pseudorandom graphs

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$. Extending classical Tur\'an type results for odd cycles, we show that $\pi_{F}(\Gamma)=1/2$ provided $F$ is an odd cycle and $\Gamma$ is a sufficiently pseudorandom graph. In particular, for $(n,d,\lambda)$-graphs $\Gamma$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-\lambda,\lambda]$, our result holds for odd cycles of length $\ell$, provided \[ \lambda^{\ell-2}\ll \frac{d^{\ell-1}}n\log(n)^{-(\ell-2)(\ell-3)}\,. \] Up to the polylog-factor this verifies a conjecture of Krivelevich, Lee, and Sudakov. For triangles the condition is best possible and was proven previously by Sudakov, Szab\'o, and Vu, who addressed the case when $F$ is a complete graph. A construction of Alon and Kahale (based on an earlier construction of Alon for triangle-free $(n,d,\lambda)$-graphs) shows that our assumption on $\Gamma$ is best possible up to the polylog-factor for every odd $\ell\geq 5$.

An Extension of Richardson’s Theorem in m-Colored Digraphs

A digraph $$D$$D is said to be an $$m$$m-coloured digraph if its arcs are coloured with $$m$$m colors. A directed path is called monochromatic if all of its arcs are coloured alike. A set $$N \subseteq $$N⊆ V ($$D$$D) of vertices of $$D$$D is said to be a kernel by monochromatic paths of the $$m$$m-coloured digraph $$D$$D if it satisfies the two following properties : (1) for any two different vertices $$x$$x, $$y$$y$$\in $$? N there is no monochromatic directed path between them, and (2) for each vertex $$u$$u$$\in $$? V($$D$$D)$$\setminus N$$\N there exists a $$uv$$uv-monochromatic directed path, for some $$v$$v$$\in N$$?N. Let $$D$$D be an arc-coloured digraph. In 2009 Galeana-Sanchez introduced the concept of color-class digraph of $$D$$D, denoted by $$\fancyscript{C}_C$$CC($$D$$D), as follows: the vertices of the color-class digraph are the colors represented in the arcs of $$D$$D, and ($$i$$i, $$j$$j) $$\in $$?$$A$$A($$\fancyscript{C}_C$$CC($$D$$D)) if and only if there exist two arcs namely ($$u$$u, $$v$$v) and ($$v$$v, $$w$$w) in $$D$$D such that ($$u$$u, $$v$$v) has color $$i$$i and ($$v$$v, $$w$$w) has color $$j$$j. Galeana-Sanchez introduced the concept of color-class digraph mainly to prove that if $$D$$D is a fnite arc-coloured digraph such that $$\fancyscript{C}_C$$CC($$D$$D) is a bipartite digraph, then D has a kernel by monochromatic paths. In this paper we will see some results related to the above result, which show the existence of kernels by monochromatic paths in arc-coloured digraphs. In particular, we will prove the following generalization of the previous result: Let $$D$$D be an $$m$$m-coloured digraph and $$\fancyscript{C}_C$$CC($$D$$D) its color-class digraph. If $$\fancyscript{C}_C$$CC($$D$$D) has no cycles of odd length at least 3, then $$D$$D has a kernel by monochromatic paths. Finally we will prove Richardson's Theorem as a direct consequence of the previous result.