Let G be a K 1;r-free graph (r 3) on n vertices. We prove that, for any induced path or induced cycle on k vertices in G (k 2r ? 1 or k 2r, respectively), the degree sum of its vertices is at most (2r ? 2)(n ?) where is the independence number of G. As a corollary we obtain an upper bound on the length of a longest induced path and a longest induced cycle in a K 1;r-free graph. Stronger bounds ...

Let G be a K4-free 4-regular connected graph with & vertices. Then G contains an independent set of vertices of cardinality at least (7&&4) 26. Furthermore, the proof will yield a polynomial-time algorithm which will return an independent set of cardinality at least (7&&4) 26 for any such graph. A family of K4 -free 4-regular connected graphs with maximum independent set of cardinality 3 11& is...

A graph G on n vertices is pancyclic if G contains cycles of all lengths`for 3 ` n and G is cycle extendable if for every nonhamiltonian cycle C G there is a cycle C 0 G such that V (C) V (C 0) and jV (C 0) n V (C)j = 1. We prove that (i) every 2-connected K 1;3-free graph is pancyclic, if G is P 5-free and n 6, if G is P 6-free and n 10, or if G is P 7-free, deer-free and n 14, and (ii) every ...

For a graph F , we say that another G is -saturated, if -free and adding any edge to would create copy of . We study for given integer n whether there exists regular -vertex -saturated graph, it does, what the smallest number edges such graph. mainly focus on case when complete prove example K 3 vertices every large enough also two relaxed versions problem: only require no supergraph should exi...

Let cl(G) denote the Ryjáček’s closure of a claw-free graph G. In this paper, we prove the following result. Let G be a 4-connected claw-free graph. Assume that G[NG(T )] is cyclically 3-connected if T is a maximal K3 in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.’s theorem [J. Graph Theory 48 (2005), no. 4, 267-276] and Pfend...

Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3, 3, 3, 1, 1, 1, 1, 1), i.e. T is a chain of three triangles. We show that every 4-connected {T,K1,3}-free graph has a hamiltonian cycle.

Let H be a family of connected graphs. A graph G is said to be H-free if G is H-free for every graph H in H. In [1] it was pointed that there is a family of connected graphs H not containing any induced subgraph of the claw having the property that the set of H-free connected graphs containing a claw is finite, provided also that those graphs have minimum degree at least two and maximum degree ...

The relationship between posets that are cycle-free and graphs that have more than one end is considered. We show that any locally finite bipartite graph corresponding to a cycle-free partial order has more than one end, by showing a correspondence between the ends of the graph and those of the Hasse graph of its Dedekind–MacNeille completion. If, in addition, the cycle-free partial order is k-...

Graph theory based methods are powerful means for representing structural systems so that their geometry and topology can be understood clearly. The combination of graph theory based methods and some metaheuristics can offer effective solutions for complex engineering optimization problems. This paper presents a Charged System Search (CSS) algorithm for the free shape optimizations of thin-wall...

There are several graphs H for which the precise structure of graphs that do not contain a minor isomorphic to H is known. In particular, such structure theorems are known for K5 [13], V8 [8] and [7], the cube [3], the octahedron [4], and several others. Such characterizations can often be very useful, e.g., Hadwiger’s conjecture for k = 4 is verified by using the structure for K5-free graphs, ...