862 research outputs found
Minimum Degree and Disjoint Cycles in Claw-Free Graphs
https://digitalcommons.memphis.edu/speccoll-faudreerj/1219/thumbnail.jp
Minimum Degree and Disjoint Cycles in Generalized Claw-Free Graphs
https://digitalcommons.memphis.edu/speccoll-faudreerj/1217/thumbnail.jp
Extremal problems on cycle structure and colorings of graphs
In this Thesis, we consider two main themes: conditions that guarantee diverse cycle structure within a graph, and the existence of strong edge-colorings for a specific family of graphs.
In Chapter 2 we consider a question closely related to the Matthews-Sumner conjecture, which states that every 4-connected claw-free graph is Hamiltonian. Since there exists an infinite family of 4-connected claw-free graphs that are not pancyclic, Gould posed the problem of characterizing the pairs of graphs, {X,Y}, such that every 4-connected {X,Y}-free graph is pancyclic. In this chapter we describe a family of pairs of graphs such that if every 4-connected {X,Y}-free graph is pancyclic, then {X,Y} is in this family. Furthermore, we show that every 4-connected {K_(1,3),N(4,1,1)}-free graph is pancyclic. This result, together with several others, completes a characterization of the family of subgraphs, F such that for all H in ∈, every 4-connected {K_(1,3), H}-free graph is pancyclic.
In Chapters and 4 we consider refinements of results on cycles and chorded cycles. In 1963, Corrádi and Hajnal proved a conjecture of Erdös, showing that every graph G on at least 3k vertices with minimum degree at least 2k contains k disjoint cycles. This result was extended by Enomoto and Wang, who independently proved that graphs on at least 3kvertices with minimum degree-sum at least 4k - 1 also contain k disjoint cycles. Both results are best possible, and recently, Kierstead, Kostochka, Molla, and Yeager characterized their sharpness examples. A chorded cycle analogue to the result of Corrádi and Hajnal was proved by Finkel, and a similar analogue to the result of Enomoto and Wang was proved by Chiba, Fujita, Gao, and Li. In Chapter 3 we characterize the sharpness examples to these statements, which provides a chorded cycle analogue to the characterization of Kierstead et al.
In Chapter 4 we consider another result of Chiba et al., which states that for all integers r and s with r + s ≥ 1, every graph G on at least 3r + 4s vertices with ẟ(G) ≥ 2r+3s contains r disjoint cycles and s disjoint chorded cycles. We provide a characterization of the sharpness examples to this result, which yields a transition between the characterization of Kierstead et al. and the main result of Chapter 3.
In Chapter 5 we move to the topic of edge-colorings, considering a variation known as strong edge-coloring. In 1990, Faudree, Gyárfás, Schelp, and Tuza posed several conjectures regarding strong edge-colorings of subcubic graphs. In particular, they conjectured that every subcubic planar graph has a strong edge-coloring using at most nine colors. We prove a slightly stronger form of this conjecture, showing that it holds for all subcubic planar loopless multigraphs
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Claw-free t-perfect graphs can be recognised in polynomial time
A graph is called t-perfect if its stable set polytope is defined by
non-negativity, edge and odd-cycle inequalities. We show that it can be decided
in polynomial time whether a given claw-free graph is t-perfect
The Cycle Spectrum of Claw-free Hamiltonian Graphs
If is a claw-free hamiltonian graph of order and maximum degree
with , then has cycles of at least many different lengths.Comment: 9 page
Packing 3-vertex paths in claw-free graphs and related topics
An L-factor of a graph G is a spanning subgraph of G whose every component is
a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the
domination number of G. A claw is a graph with four vertices and three edges
incident to the same vertex. A graph is claw-free if it has no induced subgraph
isomorphic to a claw. Our results include the following. Let G be a 3-connected
claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in
G. Then
(a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2)
if v(G) = 1 mod 3, then G - x has an L-factor, (a3) if v(G) = 2 mod 3, then G -
{x,y} has an L-factor, (a4) if v(G) = 0 mod 3 and G is either cubic or
4-connected, then G - P has an L-factor, (a5) if G is cubic with v(G) > 5 and E
is a set of three edges in G, then G - E has an L-factor if and only if the
subgraph induced by E in G is not a claw and not a triangle, (a6) if v(G) = 1
mod 3, then G - {v,e} has an L-factor for every vertex v and every edge e in G,
(a7) if v(G) = 1 mod 3, then there exist a 4-vertex path N and a claw Y in G
such that G - N and G - Y have L-factors, and (a8) d(G) < v(G)/3 +1 and if in
addition G is not a cycle and v(G) = 1 mod 3, then d(G) < v(G)/3.
We explore the relations between packing problems of a graph and its line
graph to obtain some results on different types of packings. We also discuss
relations between L-packing and domination problems as well as between induced
L-packings and the Hadwiger conjecture.
Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge
disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced
packing, graph domination, graph minor, the Hadwiger conjecture.Comment: 29 page
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