Minimally k-connected graphs of low order and maximal size

Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) $\text{e(G)?kn?(}\text{k+1}\text{2}\text{)}$; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph K_k,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove $\text{e(G)??}\text{(n+k)}{}^2\text{8}\text{?}$ and characterize all minimally k-connected graphs of order n and size $\text{?(}\text{(n+k)}{}^2\text{8}\text{?}$.     

On a conjecture of bondy

Bondy conjectured [1] that: if G is a k-connected graph, where k ≥ 2, such that the degree-sum of any k + 1 independent vertices is at least m, then G contains a cycle of length at least: $\text{Min}\text{(}\text{2m}\text{(k + 1)}\text{, n)}$ (n denotes the order of G). We prove here that this result is true.     

Uniqueness and faithfulness of embedding of toroidal graphs

A topological generalization of the uniqueness of duals of 3-connected planar graphs will be obtained. A graph G is uniquely embeddable in a surface F if for any two embeddings $\text{?}{}_1$, $\text{?}{}_2\text{:G → F}$, there are an autohomeomorphism h:F → F and an automorphism σ:G → G such that $\text{h°?}{}_1\text{= ?}{}_2\text{°σ}$. A graph G is faithfully embedabble in a surface F if there is an embedding $\text{?:G → F}$ such that for any automorphism σ:G → G, there is an autohomeomorphism h:F → F with $\text{h°? = f°σ}$. Our main theorems state that any 6-connected toroidal graph is uniquelly embeddable in a torus and that any 6-connected toroidal graph with precisely three exceptions is faithfully embeddable in a torus. The proofs are based on a classification of 6-regular torus graphs.     

Minimum number of edges in graphs that are both P2- and Pi-connected

A simple graph with n vertices is called P_i-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P₂- and P_i-connected are obtained. Namely if $\text{i?}\text{1}\text{2}\text{(n+1)}$, then |E(G)|?((4i?5)/(2i?2))(n?1), and if $\text{i >}\text{1}\text{2}\text{(n+ 1)}$, then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.     

A remark on Hamiltonian cycles

Every 2-connected graph G with $\text{δ ? (v + κ)}\text{3}$ is hamiltonian where v denotes the order, δ the minimum degree and κ the point connectivity of G.     

Some Asymptotic ith Ramsey numbers

Let G be a graph with chromatic number χ(G) and let t(G) be the minimum number of vertices in any color class among all χ(G)-vertex colorings of G. Let H′ be a connected graph and let H be a graph obtained by subdividing (adding extra vertices to) a fixed edge of H′. It is proved that if the order of H is sufficiently large, the ith Ramsey number r_i(G, H) equals $\text{[}\text{((χ(G)?1)(|}\text{H}\text{|?1)+t(G)?1)}\text{i}\text{]+1}$.     

Tough graphs and hamiltonian circuits

The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t₀, every t₀-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t₀ = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ($\text{3}\text{2}$)-tough nonhamiltonian graphs.     

A sufficient condition for hamiltonian circuits

A theorem is proved that is (in a sense) the best possible improvement on the following theme: If G is an undirected graph on n vertices in which $\text{|Γ(S)| ≥}\text{1}\text{3}\text{(n + | S | + 3)}$ for every non-empty subset S of the vertices of G, then G is Hamiltonian.     

On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

On n-hamiltonian graphs

A graph G of order p ? 3 is called n-hamiltonian, 0 ? n ? p ? 3, if the removal of any m vertices, 0 ? m ? n, results in a hamiltonian graph. A graph G of order p ? 3 is defined to be n-hamiltonian, ?p ? n ? 1, if there exists ?n or fewer pairwise disjoint paths in G which collectively span G. Various conditions in terms of n and the degrees of the vertices of a graph are shown to be sufficient for the graph to be n-hamiltonian for all possible values of n. It is also shown that if G is a graph of order p ? 3 and $\text{K}\text{(G) ? β(G) + n (?p ? n ? p ? 3)}$, then G is n-hamiltonian.     

Comparability graphs and intersection graphs

A function diagram (f-diagram) D consists of the family of curves {$\text{1}\text{?}$ñ} obtained from n continuous functions $\text{f}{}_{\mathrm{i}}\text{:[0,1]→}\text{R}\text{(1?i?n)}$. We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the f_i are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of $\text{G}\text{?}\text{is ?k+1}$. Computational complexity results are obtained for recognizing such graphs.     

On the minimum order of graphs with given semigroup

Denote by M(n) the smallest positive integer such that for every n-element monoid M there is a graph G with at most M(n) vertices such that End(G) is isomorphic to M. It is proved that $\text{2}\text{(1 + o(1))n}\text{log}{}_2\text{n}\text{≤M(n)≤n ·}\text{2n}\text{+ O(n)}$. Moreover, for almost all n-element monoids M there is a graph G with at most 12 · n · log₂n + n vertices such that End(G) is isomorphic to M.     

The binding number of a graph and its Anderson number

The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) ≥ c, then G contains at least $\text{|G|}\text{c}\text{(c + 1)}$ disjoint edges if $\text{0 ≤ c ≤}\text{1}\text{2}$, at least $\text{| G |}\text{(3c ? 2)}\text{3c}\text{?}\text{2(c ? 1)}\text{c}$ disjoint edges if $\text{1 ≤ c ≤}\text{4}\text{3}$, a Hamiltonian circuit if $\text{c ≥}\text{3}\text{2}$, and a circuit of length at least $\text{3(| G | ?1)(c ? 1)}\text{c}$ if $\text{1 < c ≤}\text{3}\text{2}$ and G is not one of two specified exceptional graphs. Each of these results is best possible.The Anderson number of a graph is defined. The Anderson numbers of a few very simple graphs are determined; and some rather weak bounds are obtained, and some conjectures made, on the Anderson numbers of graphs in general.     

On complementary graphs with no isolated vertices

Let $\text{G}$ denote the complement of a graph G, and x(G), β₁(G), β₄(G), α₀(G), α₁(G) denote respectively the chromatic number, line-independence number, point-independence number, point-covering number, line-covering number of G, Nordhaus and Gaddum showed that for any graph G of order $\text{p}\text{, {2√p} ? x(G) + x(}\text{G}\text{) ? p + 1}\text{and p}\text{? x(G)·x(}\text{G}\text{) ? [(}\text{1}\text{2}\text{(p + 1))}{}^2\text{]}$. Recently Chartrand and Schuster have given the corresponding inequalities for the independence numbers of any graph G. However, combining their result with Gallai's well known formula β₁(G) + α₁(G) = ?, one is not deduce the analogous bounds for the line-covering numbers of $\text{G}\text{and}\text{G}$, since Gallai's formula holds only if G contains no isolated vertex. The purpose of this note is to improve the results of Chartland and Schuster for line-independence numbers for graphs where both $\text{G}\text{and}\text{G}$ contain no isolated vertices and thereby allowing us to use Gallai's formula to get the corresponding bounds for the line-covering numbers of G.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

Graph equations for line graphs,total graphs,middle graphs and quasi-total graphs

Let G be a graph with vertex-set V(G) and edge-set X(G). Let L(G) and T(G) denote the line graph and total graph of G. The middle graph M(G) of G is an intersection graph Ω(F) on the vertex-set V(G) of any graph G. Let F = V′(G) ∪ X(G) where V′(G) indicates the family of all one-point subsets of the set V(G), then $\text{M(G) = Ω(F)}$.The quasi-total graph P(G) of G is a graph with vertex-set V(G)∪X(G) and two vertices are adjacent if and only if they correspond to two non-adjacent vertices of G or to two adjacent edges of G or to a vertex and an edge incident to it in G.In this paper we solve graph equations $\text{L(G) ? P(H);}\text{L(G)}\text{? P(H); P(G) ? T(H);}\text{P(G)}\text{? T(H); M(G) ? P(H);}\text{M(G)}\text{? P(H)}$.     

Hamiltonian cycles in particular k-partite graphs

Let $\text{G}$(itk, p) denote the class of k-partite graphs, where each part is a stable set of cardinality p and where the edges between any pair of stable sets are those of a perfect matching. Maru?i? has conjectured that if G belongs to $\text{G}$(k, p) and is connected then G is hamiltonian. It is proved that the conjecture is true for k ≤ 3 or p ≤ 3; but for k ≥ 4 and p ≥ 4 a non-hamiltonian connected graph in $\text{G}$(k, p) is constructed.     

Complexity of representation of graphs by set systems

Let $\text{F}$ be a family of subsets of S and let G be a graph with vertex set $\text{V={x}{}_{\mathrm{A}}\text{|A ∈}\text{F}\text{}}$ such that: (x_A, x_B) is an edge iff $\text{A?B≠}\text{0}\text{/}$. The family $\text{F}$ is called a set representation of the graph G.It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets.The set representations of random graphs are also considered.     

Relative lengths of paths and cycles in k-connected graphs

Let G be a k-connected graph where k≥3. It is shown that if G contains a path L of length l then G also contains a cycle of length at least ($\text{(2k ? 4)}\text{(3k ? 4)}$) l. This result is obtained from a constructive proof that G contains 3k² ? 7k + 4 cycles which together cover every edge of L at least 2k² ? 6k + 4 times.