Odd Girth Research Articles

Lower bounds on the independence number of certain graphs of odd girth at least seven

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233–237] proved that every connected subcubic triangle-free graph G has an independent set of order at least ( 4 n ( G ) − m ( G ) − 1 ) / 7 where n ( G ) and m ( G ) denote the order and size of G , respectively. We conjecture that every connected subcubic graph G of odd girth at least seven has an independent set of order at least ( 5 n ( G ) − m ( G ) − 1 ) / 9 and verify our conjecture under some additional technical assumptions.

A strengthening and a multipartite generalization of the Alon-Boppana-Serre theorem

The Alon-Boppana theorem confirms that for every $\epsilon>0$ and every integer $d\ge3$, there are only finitely many $d$-regular graphs whose second largest eigenvalue is at most $2\sqrt{d-1}-\epsilon$. Serre gave a strengthening showing that a positive proportion of eigenvalues of any $d$-regular graph must be bigger than $2\sqrt{d-1}-\epsilon$. We provide a multipartite version of this result. Our proofs are elementary and work also in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree $d$ and bounded maximum degree. The two-partite result shows that for every $\epsilon>0$ and any positive integers $d_1,d_2,d$, every $n$-vertex graph of maximum degree at most $d$, whose vertex set is the union of (not necessarily disjoint) subsets $V_1,V_2$, such that every vertex in $V_i$ has at least $d_i$ neighbors in $V_{3-i}$ for $i=1,2$, has $\Omega_\epsilon(n)$ eigenvalues that are larger than $\sqrt{d_1-1}+\sqrt{d_2-1}-\epsilon$. Finally, we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound $2\sqrt{d-1}-\epsilon$ can be replaced by $2\sqrt{d-1} + \delta$ for some $\delta>0$ if graphs have bounded "global girth". On the other side of the spectrum, if the odd girth is large, then we get an Alon-Boppana-Serre type theorem for the negative eigenvalues as well.

On Graphs with Cyclic Defect or Excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

K-fold coloring of planar graphs

A k-fold n-coloring of G is a mapping φ: V(G) → Zk(n) where Zk(n) is the collection of all k-subsets of {1, 2,..., n} such that φ(u) ∩ φ(v) = \( \not 0 \) if uv ∈ E(G). If G has a k-fold n-coloring, i.e., G is k-fold n-colorable. Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number, denoted by χk(G). In this paper, we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk(C*)-colorable, where C* is a shortest odd cycle of G. Moreover, we investigate that every planar graph with odd girth at least 10k − 9 (k ⩾ 3) can be k-fold (2k + 1)-colorable.

On upper bounds of odd girth cages

We give a construction of k -regular graphs of girth g using only geometrical and combinatorial properties that appear in any ( k ; g + 1 ) -cage, a minimal k -regular graph of girth g + 1 . In this construction, g ≥ 5 and k ≥ 3 are odd integers, in particular when k − 1 is a power of 2 and ( g + 1 ) ∈ { 6 , 8 , 12 } we use the structure of generalized polygons. With this construction we obtain upper bounds for the ( k ; g ) -cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g = 5 , 7 , 11 .

Online coloring graphs with high girth and high odd girth

We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high odd girth generalizing Kierstead’s algorithm for graphs that contain neither a C 3 or C 5 as an induced subgraph.

On the connectivity and superconnected graphs with small diameter

In this paper, first we prove that any graph G is 2-connected if d i a m ( G ) ≤ g − 1 for even girth g , and for odd girth g and maximum degree Δ ≤ 2 δ − 1 where δ is the minimum degree. Moreover, we prove that any graph G of diameter d i a m ( G ) ≤ g − 2 satisfies that (i) G is 5-connected for even girth g and Δ ≤ 2 δ − 5 , and (ii) G is super- κ for odd girth g and Δ ≤ 3 δ / 2 − 1 .

A construction of an infinite family of 2-arc transitive polygonal graphs of arbitrary odd girth

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C . If m is the girth of Γ then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary odd girth with 2-arc transitive automorphism groups.

On the Superconnectivity in graphs with odd girth g and even girth h

Abstract A maximally connected graph G of minimum degree δ is said to be superconnected (for short super-κ) if all disconnecting sets of cardinality δ are the neighborhood of some vertex of degree δ. Sufficient conditions on the diameter to guarantee that a graph of odd girth g and even girth h ⩾ g + 3 is super-κ are stated. Also polarity graphs are shown to be super-κ.

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.

Note on robust critical graphs with large odd girth

A graph G is ( k + 1 ) -critical if it is not k -colourable but G − e is k -colourable for any edge e ∈ E ( G ) . In this paper we show that for any integers k ≥ 3 and l ≥ 5 there exists a constant c = c ( k , l ) > 0 , such that for all n ̃ , there exists a ( k + 1 ) -critical graph G on n vertices with n > n ̃ and odd girth at least ℓ , which can be made ( k − 1 ) -colourable only by the omission of at least c n 2 edges.

Superconnectivity of regular graphs with small diameter

A graph is superconnected, for short super- κ , if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r -regular graph of diameter D and odd girth g that if D ≤ g − 2 , then the graph is super- κ when g ≥ 5 and a complete graph otherwise.

Approximating the maximum 3-edge-colorable subgraph problem

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least ( 1 − 2 3 γ o ( G ) ) of its edges, where γ o ( G ) denotes the odd girth of G . In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13 / 15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6 / 7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound. For a fixed value of a parameter k ≥ 1 , the Maximum k -Edge-Colorable Subgraph Problem asks to k -edge-color the most of the edges of a simple graph received in input. The problem is known to be A P X -hard for all k ≥ 2 . However, approximation algorithms with approximation ratios tending to 1 as k goes to infinity are also known. At present, the best known performance ratios for the cases k = 2 and k = 3 were 5 / 6 and 4 / 5 , respectively. Since the proofs of our structural result are algorithmic, we obtain an improved approximation algorithm for the case k = 3 , achieving approximation ratio of 6 / 7 . Better bounds, and allowing also for parallel edges, are obtained for graphs of higher odd girth (e.g., a bound of 13 / 15 when the input multigraph is restricted to be triangle-free, and of 19 / 21 when C 5 ’s are also banned).

Odd‐angulated graphs and cancelling factors in box products

AbstractUnder what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ϕ: G□ H→S□T is a graph homomorphism, then either ϕ maps G□{h} to S□{th} for all h∈V(H) where th∈V(T) depends on h; or ϕ maps G□{h} to {sh}□ T for all h∈V(H) where sh∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008

A note on n-tuple colourings and circular colourings of planar graphs with large odd girth

The existence of planar graphs with odd girth 2k+1 and high girth that cannot be (2k+1, k)-coloured was left as an open question by Klostermeyer and Zhang. In this note we show that such graphs exist for arbitrarily large k. We also show that these graphs have fractional chromatic number greater than 2+1/k.

Graphs with Large Girth Not Embeddable in the Sphere

In 1972, Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere Sd in such a way that two vertices joined by an edge have distance more than $\sqrt 3$(ie, distance more than 2π/3 on the sphere). In 1978, Larman [LAR] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed $\sqrt{8/3}$. In addition, he conjectured that the right answer would be $\sqrt{2}$, which is not better than the class of all graphs. Larman'sconjecture was independently proved by Rosenfeld [MR] and Rödl [VR[. In this last paper it was shown that no bound better than $\sqrt 2$ can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is stilltrue for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree.

A lower bound on the order of regular graphs with given girth pair

AbstractThe girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 (1983), 209–218]. A (δ, g)‐cage is a smallest δ‐regular graph with girth g. For all δ ≥ 3 and odd girth g ≥ 5, Harary and Kovács conjectured the existence of a (δ,g)‐cage that contains a cycle of length g + 1. In the main theorem of this article we present a lower bound on the order of a δ‐regular graph with odd girth g ≥ 5 and even girth h ≥ g + 3. We use this bound to show that every (δ,g)‐cage with δ ≥ 3 and g ∈ {5,7} contains a cycle of length g + 1, a result that can be seen as an extension of the aforementioned conjecture by Harary and Kovács for these values of δ, g. Moreover, for every odd g ≥ 5, we prove that the even girth of all (δ,g)‐cages with δ large enough is at most (3g − 3)/2. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 153–163, 2007

Connectivity of graphs with given girth pair

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209–218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then ( g , h ) is called the girth pair of the graph. In this paper we prove that a graph with girth pair ( g , h ) such that g is odd and h ⩾ g + 3 is even has high (vertex-)connectivity if its diameter is at most h - 3 . The edge version of all results is also studied.

Retracts of box products with odd‐angulated factors

AbstractLet G be a connected graph with odd girth 2κ+1. Then G is a (2κ+1)‐angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ+1)‐cycle. We prove that if G is (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+3, then any retract of the box (or Cartesian) product G□ H is S□T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ+1)‐angulated if any two vertices of G are connected by a sequence of (2κ+1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+1, then any retract of G□ H is S□T where S is a retract of G and T is a connected subgraph of H or |V(S)|=1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 (1988), 169–184]. As a corollary, we get that the core of the box product of two strongly (2κ+1)‐angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ+1)‐angulated core, then either Gn is a core for all positive integers n, or the core of Gn is G for all positive integers n. In the latter case, G is homomorphically equivalent to a normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 (1998), 867–881]. In particular, let G be a strongly (2κ+1)‐angulated core such that either G is not vertex‐transitive, or G is vertex‐transitive and any two maximum independent sets have non‐empty intersection. Then Gn is a core for any positive integer n. On the other hand, let Gi be a (2κi+1)‐angulated core for 1 ≤ i≤ n where κ1 < κ2 < … < κn. If Gi has a vertex that is fixed under any automorphism for 1 ≤ i ≤ n‐1, or Gi is vertex‐transitive such that any two maximum independent sets have non‐empty intersection for 1 ≤ i ≤ n‐1, then □i=1n Gi is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r,2r+1) □ C2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r,2r+1) □ C2l+1 is a core for r > l ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory

A case of plagiarism: Dănuţ Marcu

Yesterday I started reading the paper by Danuţ Marcu that you sent me. I liked it: the results and proofs were not great but very neat. At the same time I was surprised that he gives too few references on such an important subject, and had a feeling that I had seen something like this somewhere. So today I started checking and 15 minutes ago I discovered that the paper is a very slightly modified copy of the paper by J. B. Shearer: “The independence number of dense graphs with large odd girth” (The Electronic Journal of Combinatorics 2 (1995), http://www.combinatorics.org/). This is the first time I discovered a 100% plagiarism in math publications. Then it struck me that recently I heard about a plagiarist in a letter from a well known mathematician. I found it it was about Marcu!!! Please let the fact be well known in OR community. Marcu has to get rejected no matter where he sends his or “his” papers.