Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

A note on similar edges and edge-unique line graphs

If G is a connected graph having no vertices of degree 2 and L(G) is its line graph, two results are proven: if there exist distinct edges e and f with L(G) ? e ? L(G) ? f then there is an automorphism of L(G) mapping e to f; if $\text{G ? u ¦ G ? v}$ for any distinct vertices u, v, then $\text{L(G) ? e ¦ L(G) ? f}$ for any distinct edges e, f.     

On graphs whose least eigenvalue exceeds − 1 − √2

Let G be a graph, A(G) its adjacency matrix. We prove that, if the least eigenvalue of A(G) exceeds -1 ? √2 and every vertex of G has large valence, then the least eigenvalue is at least -2 and G is a generalized line graph.     

On n-Hamiltonian line graphs

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph L^m(G) of G is defined to be L(L^m-1(G)). A graph G of order p ≥ 3 is n-Hamiltonian, 0 ≤ n ≤ p ? 3, if the removal of any k vertices, 0 ≤ k ≤ n, results in a Hamiltonian graph. It is shown that if G is a connected graph with δ(G) ≥ 3, where δ(G) denotes the minimum degree of G, then L²(G) is (δ(G) ? 3)-Hamiltonian. Furthermore, if G is 2-connected and δ(G) ≥ 4, then L²(G) is (2δ(G) ? 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K(1, 3) and for any positive integer n, the existence of an integer k such that L^m(G) is n-Hamiltonian for every m ≥ k is exhibited. Then, for the special case n = 1, bounds on (and, in some cases, the exact value of) the smallest such integer k are determined for various classes of graphs.     

The Chromatic Index of a Graph Whose Core is a Cycle of Order at Most 13

Let G be a graph. The core of G, denoted by G _Δ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) denotes the maximum degree of G. A k -edge coloring of G is a function f : E(G) → L such that |L| = k and f (e ₁) ≠ f (e ₂) for all two adjacent edges e ₁ and e ₂ of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.     

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exists a proper edge-coloring ? of G such that ?(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ(G)≠5 and without adjacent 3-cycles, or with maximum degree Δ(G)≠5,6 and without 7-cycles, then G is edge-(Δ(G)+1)-choosable.     

A characterization of block graphs

A block graph is a graph whose blocks are cliques. For each edge e=uv of a graph G, let N_e(u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e=uv∈E(G), if x∈N_e(u) and y∈N_e(v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18-22].     

Connectivity of iterated line graphs

Let k≥0 be an integer and L^k(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L²(G))≥4δ(G)−6. We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ(L²(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.     

The structure of saturated critical blocks

A 2‐connected graph G is a critical block if G ? v is not 2‐connected for every vertex v ∈ V(G). A critical block G is a saturated critical block if G + e is not a critical block for any new edge e. The structure of all saturated critical blocks and a procedure for constructing every saturated critical block are determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 223–237, 2003     

EDGE SUBDIVISIONS OF A GIVEN GEODETIC BLOCK PRESERVING ITS PROPERTY TO BE GEODETIC

1IntroductionInthispaperweshallconsideronlyundirected2-connectedsimplegraphs,i.e,graphsthatareloopless,finite,undirectedandwithoutmultipleedges.AgraphGissaidtobegeodeticifanypairofpointsofGarejoinedbyauniquepathofshortestlength,i.e,aulliquedistancepath[1].A2-connectedgeodeticgraphiscalledageodeticblock.Agrapllisgeodeticiffeachofitsblocksisgeodetic(seeStempleandWatkins['l).Obviously,oddcycle,tree,completegrapharegeodeticgraph,wecallthemthetrivialgeodeticgraph.Nowweonlycollsidertilenontrivial…     

On k-sequential and other numbered graphs

The concept of a k-sequential graph is presented as follows. A graph G with ∣V(G)∪ E(G)∣=t is called k-sequential if there is a bijection$\text{?: V(G)∪E(G) → {k,k+1,…,t+k?1}}$ such that for each edge$\text{e}\text{?}\text{=xy}$in E(G) one has$\text{?(}\text{e}\text{?}\text{) = ∣?(x)??(y)∣}$. A graph that is 1-sequential is called simply sequential, and, in particular the author has conjectured that all trees are simply sequential. In this paper an introductory study of k-sequential graphs is made. Further, several variations on the problems of gracefully or sequentially numbering the elements of a graph are discussed.     

Length of Longest Cycles in a Graph Whose Relative Length is at Least Two

Let G be a graph. We denote p(G) and c(G) the order of a longest path and the order of a longest cycle of G, respectively. Let κ(G) be the connectivity of G, and let σ ₃(G) be the minimum degree sum of an independent set of three vertices in G. In this paper, we prove that if G is a 2-connected graph with p(G) ? c(G) ≥ 2, then either (i) c(G) ≥ σ ₃(G) ? 3 or (ii) κ(G)?=?2 and p(G) ≥ σ ₃(G) ? 1. This result implies several known results as corollaries and gives a new lower bound of the circumference.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Equimatchable factor-critical graphs

A simple graph G(X, E) is factor-critical if the induced subgraph 〈X – x〉 admits a perfect matching for every vertex x of G. It is equimatchable if every maximal matching of G is maximum. The equimatchable non-factor-critical graphs have been studied by Lesk, Plummer, and Pulleyblank. In this paper, we study the equimatchable factor-critical graphs; in particular we show that if such a graph is two-connected, it is hamiltonian.     

Domination critical graphs

A set of vertices S is said to dominate the graph G if for each v ? S, there is a vertex u ∈ S with u adjacent to v. The smallest cardinality of any such dominating set is called the domination number of G and is denoted by γ(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u ∈ V(G) with v not adjacent to u, γ(G + vu) < γ(G). Thus G is k-y-critical if γ(G) = k and for each edge e ? E(G), γ(G + e) = k ?1. The 2-domination critical graphs are characterized the properties of the k-critical graphs with k ≥ 3 are studied. In particular, the connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

On the spanning connectivity of graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if it contains all vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. The spanning connectivity of G, κ^*(G), is defined to be the largest integer k such that G is w^*-connected for all 1?w?k if G is a 1^*-connected graph. In this paper, we prove that κ^*(G)?2δ(G)-n(G)+2 if (n(G)/2)+1?δ(G)?n(G)-2. Furthermore, we prove that κ^*(G-T)?2δ(G)-n(G)+2-|T| if T is a vertex subset with |T|?2δ(G)-n(G)-1.     

Clique graphs of time graphs

A simple, finite graph G is called a time graph (equivalently, an indifference graph) if there is an injective real function f on the vertices v(G) such that v_iv_j ∈ e(G) for v_i ≠ v_j if and only if |f(v_i) ? f(v_j)| ≤ 1. A clique of a graph G is a maximal complete subgraph of G. The clique graph K(G) of a graph G is the intersection graph of the cliques of G. It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K²(G), and inductively, denote K(K^m?1(G)) by K^m(G). Define the index indx(G) of a connected time graph G as the smallest integer n such that Kⁿ(G) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.     

On a construction of critical graphs which are not sensitive

Let ?(G) be theclosed-set lattice of a graphG. G issensitive if the following implication is always true for any graphG′: ?(G)??(G′)?(G)?G′G iscritical if ?(G)??(G-e) for anye inE(G) and ?(G)??(G+e) for anye in \(\left( {\bar G} \right)\) where \(\bar G\) is the complement ofG. Every sensitive graph is, a fortiori, critical. Is every critical graph sensitive? A negative answer to this question is given in this note.