26,019 research outputs found
On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions
In contrast with Kotzig's result that the line graph of a -regular graph
is Hamilton decomposable if and only if is Hamiltonian, we show that
for each integer there exists a simple non-Hamiltonian -regular
graph whose line graph has a Hamilton decomposition. We also answer a question
of Jackson by showing that for each integer there exists a simple
connected -regular graph with no separating transitions whose line graph has
no Hamilton decomposition
Reverse Line Graph Construction: The Matrix Relabeling Algorithm MARINLINGA Versus Roussopoulos's Algorithm
We propose a new algorithm MARINLINGA for reverse line graph computation,
i.e., constructing the original graph from a given line graph. Based on the
completely new and simpler principle of link relabeling and endnode
recognition, MARINLINGA does not rely on Whitney's theorem while all previous
algorithms do. MARINLINGA has a worst case complexity of O(N^2), where N
denotes the number of nodes of the line graph. We demonstrate that MARINLINGA
is more time-efficient compared to Roussopoulos's algorithm, which is
well-known for its efficiency.Comment: 30 pages, 24 figure
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
On the Representability of Line Graphs
A graph G=(V,E) is representable if there exists a word W over the alphabet V
such that letters x and y alternate in W if and only if (x,y) is in E for each
x not equal to y. The motivation to study representable graphs came from
algebra, but this subject is interesting from graph theoretical, computer
science, and combinatorics on words points of view. In this paper, we prove
that for n greater than 3, the line graph of an n-wheel is non-representable.
This not only provides a new construction of non-representable graphs, but also
answers an open question on representability of the line graph of the 5-wheel,
the minimal non-representable graph. Moreover, we show that for n greater than
4, the line graph of the complete graph is also non-representable. We then use
these facts to prove that given a graph G which is not a cycle, a path or a
claw graph, the graph obtained by taking the line graph of G k-times is
guaranteed to be non-representable for k greater than 3.Comment: 10 pages, 5 figure
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