A 2-matching in an undirected graph G = (V G,EG) is a function x : EG → {0, 1, 2} such that for each node v ∈ V G the sum of values x(e) on all edges e incident to v does not exceed 2. The size of x is the sum ∑ e x(e). If {e ∈ EG | x(e) 6= 0} contains no triangles then x is called triangle-free. Cornuéjols and Pulleyblank devised a combinatorial O(mn)-algorithm that finds a triangle free 2-mat...

The chain graph sandwich problem asks: Given a vertex set V , a mandatory edge set E1, and a larger edge set E2, is there a graph G= (V ,E) such that E1 ⊆ E ⊆ E2 with G being a chain graph (i.e., a 2K2-free bipartite graph)? We prove that the chain graph sandwich problem is NP-complete. This result stands in contrast to (1) the case where E1 is a connected graph, which has a linear-time solutio...

Seymour conjectured for a fixed integer k ≥ 2 that if G is a graph of order n with δ(G) ≥ kn/(k + 1), then G contains the kth power Ck n of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k = 2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s ≥ 3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. S...

The Coloring problem is to test whether a given graph can be colored with at most k colors for some given k, such that no two adjacent vertices receive the same color. The complexity of this problem on graphs that do not contain some graph H as an induced subgraph is known for each fixed graph H. A natural variant is to forbid a graph H only as a subgraph. We call such graphs strongly H-free an...

In a graph G of maximum degree 3, let γ(G) denote the largest fraction of edges that can be 3 edge-coloured. Rizzi [9] showed that γ(G) ≥ 1 − 2 3godd(G) where godd(G) is the odd girth of G, when G is triangle-free. In [3] we extended that result to graph with maximum degree 3. We show here that γ(G) ≥ 1− 2 3godd(G)+2 , which leads to γ(G) ≥ 15 17 when considering graphs with odd girth at least ...

Recently, research on parallel processing systems is very active, and many complex topologies have been proposed. A burnt pancake graph is one such topology. In this paper, we prove that a faulty burnt pancake graph with degree n has a fault-free Hamiltonian cycle if the number of faulty elements is no more than n − 2, and it has a fault-free Hamiltonian path between any pair of non-faulty node...

Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C), where C is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass-free line graph is ...

In this paper we study the chromatic number for graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P 5-free and triangle-free, P 6-free and C 6-free graphs are 3-colourable. A canonical extension of these graph classes is G

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [J. of Combinatorial Theory, Ser. B. 82 (2001), 306-315] that every 4-connected line graph of a claw free graph...

A graph is called 2K2-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every 3 2 -tough 2K2-free graph with at least three vertices has a spanning trail with maximum degree at most 4. In this paper, we confirm this conjecture. We also provide examples for all t < 54 of t-tough graphs that do not have a spanning trail with maximum degre...