22 research outputs found
Toughness and hamiltonicity in -trees
We consider toughness conditions that guarantee the existence of a hamiltonian cycle in -trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to . It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to -trees for : Let be a -tree. If has toughness at least then is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough -trees for each $k\ge 3
Uniquely List Colorability of Complete Split Graphs
The join of null graph Om and complete graph Kn, denoted by S(m; n), is called a complete split graph. In this paper, we characterize unique list colorability of the graph G = S(m; n). We shall prove that G is uniquely 3-list colorable graph if and only if m>=4, n>=4 and m + n>=10, m(G)>=4 for every 1<=m<=5 and n>=6.The join of null graph Om and complete graph Kn, denoted by S(m; n), is called a complete split graph. In this paper, we characterize unique list colorability of the graph G = S(m; n). We shall prove that G is uniquely 3-list colorable graph if and only if m>=4, n>=4 and m + n>=10, m(G)>=4 for every 1<=m<=5 and n>=6
Minimally toughness in special graph classes
Let be a positive real number. A graph is called -tough, if the
removal of any cutset leaves at most components. The toughness of a
graph is the largest for which the graph is -tough. A graph is minimally
-tough, if the toughness of the graph is and the deletion of any edge
from the graph decreases the toughness. In this paper we investigate the
minimum degree and the recognizability of minimally -tough graphs in the
class of chordal graphs, split graphs, claw-free graphs and -free graphs
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