Hamilton cycles in 5-connected line graphs

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness

Pancyclicity of Hamiltonian line graphs

Let f(n) be the smallest integer such that for every graph G of order n with minimum degree 3(G)>f(n), the line graph L(G) of G is pancyclic whenever L(G) is hamiltonian. Results are proved showing that f(n) = ®(n 1/3)

Conjecture jackson dalam subgrapheuler

ABSTRAK Conjecture Jackson menyatakan bahwa jika G .merupakan graph 2-garis terhubung, maka G mempunyai subgraph euler H dengan IV(H)I ?. 2, sedemikian sehingga untuk setiap komponen F dari G-V(H) terdapat aebanyak-banyaknya 3 garis antara F dan H. Conjecture Thomassen menyatakan bahwa semua line graph 4-terhubung merupakan hamiltonian. Dalam skripsi ini ditunjukkan hubungan antara Conjecture Jackson dan Conjecture Thomassen serta eksistensi dari J-subgraph dan Line graph hamiltonian. ABSTRACT Jackson conjectured that if G is a 2-edge-connected graph, then G has an eulerian subgraph H with 1V(H)1 2 such that for each component F of G-V(H), there are at most three edges between F and H. Thomassen conjectured that all 4-connected line graphs are hamiltonian. In this research be showed the connectivity between Jackson's conjecture and Thomassen's conjecture, and the existence of J-subgraphs and hamiltonian line graphs

A Survey of Line Graphs and Hamiltonian Paths

In this paper, we are going to explore a survey of line graphs and hamiltonian paths. Research concerning line graphs and hamiltonian paths started in the 1960\u27s. We will investigate some recent theorems and proofs covering this topic. At the end, we will prove a main result involving line graphs and hamiltonian paths

A result on Hamiltonian line graphs involving restrictions on induced subgraphs

It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph

Group Colorability and Hamiltonian Properties of Graphs

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs by proving an analogue to Brooks\u27 Theorem

3-Connected line graphs of triangular graphs are panconnected and 1-hamiltonian

A graph is k-triangular if each edge is in at least k triangles. Triangular is a synonym for 1-triangular. It is shown that the line graph of a triangular graph of order at least 4 is panconnected if and only if it is 3-connected. Furthermore, the line graph of a k-triangular graph is k-hamiltonian if and only if it is (k + 2)-connected (k ≥ 1). These results generalize work of Clark and Wormald and of Lesniak-Foster. Related results are due to Oberly and Sumner and to Kanetkar and Rao