Odd Cycle Of Length Research Articles

Tight bound for matching

Let M be the number of edges in a maximum matching in graphs with m edges, maximum vertex degree k and shortest simple odd-length cycle length L. We show that $$M\geq \left \{\begin{array}{l@{\quad }l}\frac{m}{2}-\frac{m}{2L},&\mbox{if}\ k=2,\\\noalign{\vspace{2pt}}\frac{m}{k}-\frac{m}{(k+L)k},&\mbox{if}\ k>2.\end{array}\right.$$ This lower bound is tight. When no simple odd-length cycle exists it is known previously that $M\geq \frac{m}{k}$.

Associated primes of monomial ideals and odd holes in graphs

Let G be a finite simple graph with edge ideal I(G). Let I(G)∨ denote the Alexander dual of I(G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I(G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of (I(G)∨)2.

Cycle length parities and the chromatic number

AbstractIn 1966 Erdös and Hajnal proved that the chromatic number of graphs whose odd cycles have lengths at most l is at most l+1. Similarly, in 1992 Gyárfás proved that the chromatic number of graphs which have at most k odd cycle lengths is at most 2k+2 which was originally conjectured by Bollobás and Erdös. Here we consider the influence of the parities of the cycle lengths modulo some odd prime on the chromatic number of graphs. As our main result we prove the following: Let p be an odd prime, k∈ℕ and I⊆{0, 1, …, p−1} with |I|≤p−1. If G is a graph such that the set of cycle lengths of G contains at most k elements which are not in I modulo p, then χ(G)≤(1+|I|/(p−|I|))k+p(p−1)(r(2p, 2p)+1)+1 where r(p, q) denotes the ordinary Ramsey number. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 210–218, 2010

Graphs of odd girth 7 with large degree

Abstract We show that every graph with minimum degree δ > 4 n / 17 and no odd cycles of length 3 or 5 is homomorphic with the Mobius ladder with 6 rungs and include the extremal graph characterization in the case of equality. The key tools used in our observations are simple characteristics of maximal odd girth 7 graphs.

Hypergraphs with no odd cycle of given length

Abstract We give upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In particular, we show that a 3-uniform hypergraph containing no cycle of length 2 k + 1 has less than 4 k 4 n 1 + 1 / k + O ( n ) edges. Constructions show that these bounds are best possible (up to constant factor) for k = 1 , 2 , 3 , 5 .

Subcomplexes of box complexes of graphs

The box complex ${\sf B}(G)$ of a graph $G$ is a simplicial $\mathbf{Z}_2$-complex defined by J. Matoušek and G.M. Ziegler in \cite{MZ04}. They proved that $\chi (G)\geq \text{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| )+2$, where $\chi (G)$ is the chromatic number of $G$ and $\text{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| )$ is the $\mathbf{Z}_2$-index of ${\sf B}(G)$. In this paper, to study topology of box complexes, for the union $G\cup H$ of two graphs $G$ and $H$, we compare ${\sf B}(G\cup H)$ with its subcomplex ${\sf B}(G)\cup {\sf B}(H)$. We give a sufficient condition on $G$ and $H$ so that ${\sf B}(G\cup H)={\sf B}(G)\cup {\sf B}(H)$ and ${\sf B}(G\cap H)={\sf B}(G)\cap {\sf B}(H)$ hold. Moreover, under that condition, we show $$\rm{max} \{\chi (G), \chi (H)\}\leq \chi (G\cup H)\leq \max \{\chi (G)+l_H, \chi (H)\},$$ where $l_H$ is the number defined in Definition 3.8. Also we prove $$\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G\cup H)\| )=\max \{\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| ),\, \rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(H)\| )\,\}$$ if $\max \{\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(G)\| ),\,\rm{ind}_{\mathbf{Z}_2}(\| {\sf B}(H)\| )\,\}\geq 1$. The complex $\mathsf{B}(G)$ of a graph $G$ contains a 1-dimensional free $\mathbf{Z}_2$-subcomplex $\overline{G}$ of ${\sf B}(G)$, defined in [2]. As a supplement to [2], we show that for a connected graph $G$, $\mathsf {B}(G)$ is disconnected if and only if $\overline{G}$ is disconnected if and only if $G$ contains no cycles of odd length, or equivalently, $G$ is bipartite.

Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack

The problems that arise from the presence of self-attacking arguments and odd-length cycles of attack within argumentation frameworks are widely recognized in the literature on defeasible argumentation. This paper introduces two simple semantics to capture different intuitions about what kinds of arguments should become justified in such scenarios. These semantics are modeled upon two extensions of argumentation frameworks, which we call sustainable and tolerant. Each one is constructed on the common ground of the powerful concept of admissibility introduced by Dung in [P.M. Dung, On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming, and n-person games, Artificial Intelligence 77 (1995) 321–357]. The novelty of this approach consists in viewing the admissibility of a subset of arguments as relative to potentially challenging subsets of arguments. Both sustainable and tolerant semantics are more credulous than preferred semantics (i.e. they justify at least the same arguments, and possibly more). Given certain sufficient conditions they coincide among them as well as with other semantics introduced by Dung.

Colourings of graphs with two consecutive odd cycle lengths

In 1992 Gyárfás showed that a graph G having only k odd cycle lengths is ( 2 k + 1 ) -colourable, if it does not contain a K 2 k + 2 . In this paper, we will present the results for graphs containing only odd cycles of length 2 m − 1 and 2 m + 1 as done in [S. Matos Camacho, Colourings of graph with prescribed cycle lengths, diploma thesis, TU Bergakademie Freiberg, 2006. [3]]. We will show that these graphs are 4-colourable.

Cycle lengths in sparse graphs

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. Erdős [8] conjectured that |C(G)|=Ω ` d (g−1)/2 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 (g−1)/2 . 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 [7,11,21,24,25]. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that Ω ` d (g−1)/2 is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d.

Weighted restricted 2-matching

A perfect 2-matching of a graph is a vector assigning values 0,1, or 2 to the edges such that the sum of values of edges incident with any node is equal to 2. For restricted perfect 2-matchings, we are also given a collection of “allowed” odd cycles, and restrict ourselves to those perfect 2-matchings the support of which contains no odd cycle not in this collection. Given a graph and a collection of allowed odd cycles, we provide a TDI description of the convex hull of restricted perfect 2-matchings. The description has large coefficients and is given implicitly, thus polynomial time separation or optimization is not straightforward. In order to have such an algorithm, one has to specify the collection of allowed odd cycles. For any fixed number k, we solve optimization in strongly polynomial time for the special cases when the collection consists of odd cycles of length less than k, or odd cycles of length more than k. These solved special cases include minimum weight perfect matching, minimum weight triangle-free and/or pentagon-free perfect perfect 2-matching, and a bunch of other relaxations of the travelling salesman problem. Our algorithm is based on a primal–dual approach, and the unweighted algorithm of Cornuejols and Hartvigsen, which is used as a subroutine. The TDI description also may be regarded as a generalization of their unweighted min–max formula.

Equipartite Gregarious 5-Cycle Systems and Other Results

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph Km(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).

A bound on the chromatic number using the longest odd cycle length

Let G be a non-bipartite graph with 𝓵 as the length of the longest odd cycle. Erds and Hajnal proved that (G) 𝓵 + 1. We show that the only graphs for which this is tight are those that contain K𝓵 +...

A bound on the chromatic number using the longest odd cycle length

AbstractLet G be a non‐bipartite graph with ℓ as the length of the longest odd cycle. Erdös and Hajnal proved that χ(G) ≤ ℓ + 1. We show that the only graphs for which this is tight are those that contain Kℓ + 1 and further, if G does not contain Kℓ then χ(G) ≤ ℓ −1. We then extend these results and show that if we exclude a large clique, we can prove better bounds for χ(G) as a function of ℓ. Specifically, we show that if ω(G) ≤ ⌈ℓ(1 − ϵ)⌉ for ϵ ≤ 4/15, then χ(G) ≤ ⌈ℓ(1 − ϵ/2)⌉. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 267–276, 2007

The strong perfect graph theorem

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The ?strong perfect graph conjecture? (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu?ejols and Vuiskovi?c ? that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge?s conjecture cannot have either of these properties). In this paper we prove both of these conjectures.

Odd Hole Recognition in Graphs of Bounded Clique Size

In a graph G, an odd hole is an induced odd cycle of length at least 5. A clique of G is a set of pairwise adjacent vertices. In this paper we consider the class ${\cal C}_k$ of graphs whose cliques have a size bounded by a constant k. Given a graph G in ${\cal C}_k$, we show how to recognize in polynomial time whether G contains an odd hole.

Coloring Meyniel graphs in linear time

A Meyniel graph is a graph in which every odd cycle of length at least five has two chords. We present a linear-time algorithm that colors optimally the vertices of a Meyniel graph and finds a clique of maximum size.

Recognizing Berge Graphs

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|9). This is independent of the recent proof of the strong perfect graph conjecture.

Canonical varieties with no canonical axiomatisation

We give a simple example of a varietyV\mathbf {V}of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the varietyRRA\mathbf {RRA}of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences. Using probabilistic methods of Erdős, we construct an infinite sequenceG0,G1,…G_0,G_1,\ldotsof finite graphs with arbitrarily large chromatic number, such that eachGnG_nis a bounded morphic image ofGn+1G_{n+1}and has no odd cycles of length at mostnn. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from theGnG_nsatisfies arbitrarily many axioms from a certain axiomatisation ofV (RRA)\mathbf {V}\ (\mathbf {RRA}), while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish thatV (RRA)\mathbf {V}\ (\mathbf {RRA})has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.

Cycle lengths and chromatic number of graphs

For a simple finite graph G let C o( G) and C e( G) denote the set of odd cycle lengths and even cycle lengths in a graph G, respectively. We will show that the chromatic number χ( G) of G satisfies: χ( G)⩽ min{2 r+2,2 s+3}⩽ r+ s+2, if | C o( G)|= r and | C e( G)|= s.

On the lengths of symmetry breaking–preserving games on graphs

Given a graph G, we consider a game where two players, A and B , alternatingly color edges of G in red and in blue, respectively. Let L sym( G) be the maximum number of moves in which B is able to keep the red and the blue subgraphs isomorphic, if A plays optimally to destroy the isomorphism. This value is a lower bound for the duration of any avoidance game on G under the assumption that B plays optimally. We prove that if G is a path or a cycle of odd length n, then Ω( log n)⩽L sym (G)⩽ O( log 2 n) . The lower bound is based on relations with Ehrenfeucht–Fraı̈ssé games from model theory. We also consider complete graphs and prove that L sym( K n )=O(1).