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

The 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 ( 1 ), 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     

The face pair of planar graphs

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24-26] have introduced a generalization of the standard cage question—r-regular graphs with given odd and even girth pair. The pair (ω,ε) is the girth pair of graph G if the shortest odd and even cycles of G have lengths ω and ε, respectively, and denote the number of vertices in the (r,ω,ε)-cage by f(r,ω,ε). Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145-153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f(3,ω,4)=2ω and the bounds for the left cases. In this paper, we show the values of f(r,ω,ε) for the left cases where (r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε), (3,5,ε)}.     

Extremal Regular Graphs with Prescribed Odd Girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ƒ(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ƒ(k, g) ≥ kg/2 and that ƒ(k, g) = kg/2 if k is even. The exact values of ƒ(k, g) are also known if k = 3 or g = 5. Let x_e denote the smallest even integer no less than x, δ(g) = (−1)^{g − 1}/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ≥ 5 and g ≥ 7 are both odd, then [formula] with the exception that ƒ(5, 7) = 20.     

On the connectivity and superconnected graphs with small diameter

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

The maximum valency of regular graphs with given order and odd girth

The odd girth of a graph G is the length of a shortest odd cycle in G. Let d(n, g) denote the largest k such that there exists a k-regular graph of order n and odd girth g. It is shown that dn, g ≥ 2|n/g≥ if n ≥ 2g. As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2j-regular graph of order n and odd girth g if and only if n ≥ gj, where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.     

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let U _n,g be the lollipop graph obtained by appending a pendent vertex of a path on n ? g (n > g) vertices to a vertex of a cycle on g ? 3 vertices. We prove that the graph U _n,g uniquely minimizes the Laplacian spectral radius for n ? 2g ? 1 when g is even and for n ? 3g ? 1 when g is odd.     

Cayley Cages

A (k,g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k,g)-Cayley graphs for all parameters k≥2 and g≥3 and generalize this construction to show that many well-known small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k,g)-Cayley cages.     

On Superconnectivity of (4, g)-Cages

A (k, g)-cage is a graph that has the least number of vertices among all k-regular graphs with girth g. It has been conjectured (Fu et?al. in J. Graph Theory, 24:187?C191, 1997) that all (k, g)-cages are k-connected for every k??? 3. A k-connected graph G is called superconnected if every k-cutset S is the neighborhood of some vertex. Moreover, if G?S has precisely two components, then G is called tightly superconnected. In this paper, we prove that every (4, g)-cage is tightly superconnected when g ???11 is odd.     

Extremal non-bipartite regular graphs of girth 4

Let f(r) denote the smallest number of points in a non-bipartite r-regular graph of girth 4. It is known that $\text{f(r) ≥}\text{5r}\text{2}$ and that $\text{f(r) =}\text{5r}\text{2}$ if r is even. It is proved that $\text{f(r) ～}\text{5r}\text{2}$ and exact values for f(r) are provided for odd integers of the form r = 4n ? 1. Tight bounds for f(r) for odd integers of the form r= 4n + 1 are given.     

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.     

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n^1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n^1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n.     

Smallest regular graphs with prescribed odd girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let f(k,g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. The exact values of f(k,g) are determined if one of the following holds:

• (i) k > 2g ?5 and k is a prime number,
• (ii) k > (2?(g + 1)/4? ?1)², and
• (iiii) k is a perfect square.

     

New improvements on connectivity of cages

A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1)² ≤ δ + $ \sqrt \delta $ \sqrt \delta − 2 < r ² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta $ \frac{{r\left( {r - 1} \right)^2 }} {4} + 1 + 2r\left( {r - 1} \right) \leqslant \delta . These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.     

Noncrossing Hamiltonian paths in geometric graphs

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that . We also establish several results related to special classes of geometric graphs. Let h₁(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h₁(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that . Let h₂(n) denote the largest number such that when we remove an arbitrary star with at most h₂(n) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h₂(n)=⌈n/2⌉-1. Further we prove that when we remove any matching from a complete geometric graph the resulting graph will have a noncrossing Hamiltonian path.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

All (k;g)‐cages are k‐edge‐connected

A (k;g)‐cage is a k‐regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)‐cages are k‐edge‐connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)‐cages are k‐edge‐connected if g is odd. Combining our results, we conclude that the (k;g)‐cages are k‐edge‐connected. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 219–227, 2005