On the λ′-optimality in graphs with odd girth g and even girth h (original) (raw)

Sufficient conditions for λ′‐optimality in graphs with girth g

2006

For a connected graph the restricted edge-connectivity λ (G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in }, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ -optimality in graphs of diameter 2, Discrete Math 283 , 113-120] gave a sufficient condition for λ -optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g − 1, g being the girth of the graph, and show that a graph G with diameter at most g − 2 is λ -optimal.

Sufficient conditions for lambda'-optimality in graphs with girth g

Journal of Graph Theory, 2006

For a connected graph the restricted edge-connectivity λ′(G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in G–S. A graph G is said to be λ′-optimal if λ′(G) = ξ(G), where ξ(G) is the minimum edge-degree in G defined as ξ(G) = min{d(u) + d(v) − 2:uv ∈ E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ′-optimality in graphs of diameter 2, Discrete Math 283 (2004), 113–120] gave a sufficient condition for λ′-optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g − 1, g being the girth of the graph, and show that a graph G with diameter at most g − 2 is λ′-optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 73–86, 2006

Sufficient conditions for λ′-optimality of graphs with small conditional diameter

2005

A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edgeconnectivity λ (G) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ -optimal if λ (G) = ξ(G), where ξ(G) denotes the minimum edge-degree of G defined as ξ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. The P-diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D 1 (resp. D 2 ) be the P-diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is λ -optimal if D 1 g − 2 and D 2 g − 5. For even girth we obtain a similar result. Second, let F ⊂ V (G) with |F | = δ − 1, δ 2, being the minimum degree of G. Using the property Q of being vertices of G − F we prove that a graph with girth g / ∈ {4, 6, 8} is λ -optimal if this Q-diameter is at most 2 (g − 3)/2 .

Connectivity of graphs with given girth pair

Discrete Mathematics, 2007

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.

The minimum size of graphs satisfying cut conditions

Discrete Applied Mathematics, 2018

A graph G of order n satisfies the cut condition (CC) if there are at least |A| edges between any set A ⊂ V (G), |A| ≤ n/2, and its complement A = V (G) \ A. For even n, G satisfies the even cut condition (ECC), if [A, A] contains at least n/2 edges, for every A ⊂ V (G), |A| = n/2. We investigate here the minimum number of edges in a graph G satisfying CC or ECC. A simple counting argument shows that for both cut conditions |E(G)| ≥ n − 1, and the star K 1,n−1 is extremal. Faudree et al. (1999) conjectured that the extremal graphs with maximum degree ∆(G) < n − 1 satisfying ECC have 3n/2 − O(1) edges. Here we prove the tight bound |E(G)| ≥ 3n/2−3, for every graph G with ∆(G) < n−1 and satisfying CC. If G is 2-connected and satisfies ECC, we prove that |E(G)| ≥ 3n/2 − 2 holds and tight, for every even n. We obtain the weaker bound |E(G)| ≥ 5n/4 − 2, for every graph of order n ≡ 0 (mod 4) with ∆(G) < n − 1 and satisfying ECC; meanwhile we conjecture that |E(G)| ≥ 3n/2 − 4 holds, for every even n.

On the connectivity of certain graphs of high girth

Discrete Mathematics, 2004

Let q be a prime power and k ≥ 2 be an integer. In [2] and [3] it was determined that the number of components of certain graphs D(k, q) introduced in [1] is at least q t−1 where t = k+2 4. This implied that these components (most often) provide the best-known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. In [4], it was shown that the number of components is (exactly) q t−1 for q odd, but the method used there failed for q even. In this paper we prove that the number of components of D(k, q) for even q > 4 is again q t−1 where t = k+2 4. Our proof is independent of the parity of q as long as q > 4. Furthermore, we show that for q = 4 and k ≥ 4, the number of components is q t .

Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

2008

For a connected graph G, the rth extraconnectivity r (G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r + 1 vertices. The standard connectivity and superconnectivity correspond to 0 (G) and 1 (G), respectively. The minimum r-tree degree of G, denoted by r (G), is the minimum cardinality of N(T ) taken over all trees T ⊆ G of order |V (T )| = r + 1, N(T ) being the set of vertices not in T that are neighbors of some vertex of T. When r = 1, any such considered tree is just an edge of G. Then, 1 (G) is equal to the so-called minimum edge-degree of G, defined as (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short r -optimal, if r (G) r (G). In this paper, we present some sufficient conditions that guarantee r (G) r (G) for r 2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r = 1.

On the Girth of Graphs Critical with Respect to Edge-Colourings

Bulletin of the London Mathematical Society, 1976

A graph G with maximum valency r is called critical if r + 1 colours are needed for an edgecolouring, but every proper subgraph requires at most r. In this note we consider the minimum order/(r, g) of a critical graph of maximum valency r and girth g. We show that fir, 3) = r+1 or r+2 according as r is even or odd,/(r, 4) = 2r+1,/

On the restricted connectivity and superconnectivity in graphs with given girth

Discrete Mathematics, 2007

The restricted connectivity κ′(G)κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex uu has all its neighbors in X; the superconnectivity κ1(G)κ1(G) is defined similarly, this time considering only vertices uu in G-XG-X, hence κ1(G)⩽κ′(G)κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u)d(u) standing for the degree of a vertex uu. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(G)κ1(G)⩾ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17–31] and guaranteeing κ1(G)=κ′(G)=ξ(G)κ1(G)=κ′(G)=ξ(G) under some additional constraints.

A graph and its complement with specified properties III: girth and circumference

International Journal of Mathematics and Mathematical Sciences, 1979

Dedicated to Karl Menger ABSTRACT. We investigate the conditions under which both a graph G and its complement G possess a specified property. In particular, we characterize all graphs G for which G and G both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerlan. The proofs are elementary but amusing. KEF WORDS AND PHRASES. Graphs, Complement. AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES. 05C99.