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

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved

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, $t$-tough graphs, and claw-free graphs

On factors of 4-connected claw-free graphs

We consider the existence of several different kinds of factors in 4-connected claw-free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4-connected line graph is Hamiltonian, i.e. has a connected 2-factor. Conjecture 2 (Matthews and Sumner): Every 4-connected claw-free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass-free graphs, i.e. graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjecture 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths. \u

Heavy subgraphs, stability and hamiltonicity

Let $G$ be a graph. Adopting the terminology of Broersma et al. and \v{C}ada, respectively, we say that $G$ is 2-heavy if every induced claw ($K_{1,3}$) of $G$ contains two end-vertices each one has degree at least $|V(G)|/2$; and $G$ is o-heavy if every induced claw of $G$ contains two end-vertices with degree sum at least $|V(G)|$ in $G$. In this paper, we introduce a new concept, and say that $G$ is \emph{$S$-c-heavy} if for a given graph $S$ and every induced subgraph $G'$ of $G$ isomorphic to $S$ and every maximal clique $C$ of $G'$, every non-trivial component of $G'-C$ contains a vertex of degree at least $|V(G)|/2$ in $G$. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and $N$-c-heavy graph is hamiltonian, where $N$ is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs $S$ such that every 2-connected o-heavy and $S$-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.Comment: 21 pages, 6 figures, finial version for publication in Discussiones Mathematicae Graph Theor

Degree and neighborhood conditions for hamiltonicity of claw-free graphs

For a graph H , let σ t ( H ) = min { Σ i = 1 t d H ( v i ) | { v 1 , v 2 , … , v t } is an independent set in H } and let U t ( H ) = min { | ⋃ i = 1 t N H ( v i ) | | { v 1 , v 2 , ⋯ , v t } is an independent set in H } . We show that for a given number ϵ and given integers p ≥ t \u3e 0 , k ∈ { 2 , 3 } and N = N ( p , ϵ ) , if H is a k -connected claw-free graph of order n \u3e N with δ ( H ) ≥ 3 and its Ryjác̆ek’s closure c l ( H ) = L ( G ) , and if d t ( H ) ≥ t ( n + ϵ ) ∕ p where d t ( H ) ∈ { σ t ( H ) , U t ( H ) } , then either H is Hamiltonian or G , the preimage of L ( G ) , can be contracted to a k -edge-connected K 3 -free graph of order at most max { 4 p − 5 , 2 p + 1 } and without spanning closed trails. As applications, we prove the following for such graphs H of order n with n sufficiently large: (i) If k = 2 , δ ( H ) ≥ 3 , and for a given t ( 1 ≤ t ≤ 4 ), then either H is Hamiltonian or c l ( H ) = L ( G ) where G is a graph obtained from K 2 , 3 by replacing each of the degree 2 vertices by a K 1 , s ( s ≥ 1 ). When t = 4 and d t ( H ) = σ 4 ( H ) , this proves a conjecture in Frydrych (2001). (ii) If k = 3 , δ ( H ) ≥ 24 , and for a given t ( 1 ≤ t ≤ 10 ) d t ( H ) \u3e t ( n + 5 ) 10 , then H is Hamiltonian. These bounds on d t ( H ) in (i) and (ii) are sharp. It unifies and improves several prior results on conditions involved σ t and U t for the hamiltonicity of claw-free graphs. Since the number of graphs of orders at most max { 4 p − 5 , 2 p + 1 } are fixed for given p , improvements to (i) or (ii) by increasing the value of p are possible with the help of a computer

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