A graph is (k1, k2)-colorable if its vertex set can be partitioned into a graph with maximum degree at most k1 and and a graph with maximum degree at most k2. We show that every (C3, C4, C6)-free planar graph is (0, 6)-colorable. We also show that deciding whether a (C3, C4, C6)-free planar graph is (0, 3)-colorable is NP-complete.

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The `-Coloring problem is the problem to decide whether a graph can be colored with at most ` colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8free graphs. This improves a result of Le, Randerath, and Schierme...

In this paper, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let Zi be the graph obtaind from a triangle by attaching a path of length i to one of its vertices, and let Q∗ be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Lai et al. [J. Graph Theory 64 (2010), no. 1, 1-11] conjectured that every 3-connected {K1,3, Z9}-f...

Let e(G) denote the number of edges in a graph G, and let Ck denote a k-cycle. It is well-known that every graph has a bipartite subgraph with at least half as many edges. Győri showed that any bipartite, C6-free graph contains a C4-free subgraph containing at least half as many edges. Applying these two results in sequence we see that every C6-free graph, G, has a bipartite C4-free subgraph, H...

it is very important for managers, investors and financial policy-makers to detect and analyze factors affecting financial markets to obtain optimal decision and reduce risks. the importance of market analysis and attempt to improve its behavior understanding, has led analysts to use the experiences of other professionals in the fields such as social sciences and mathematics to examine the inte...

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v1, . . . , vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graph...

In this paper, two results about the signless Laplacian spectral radius q(G) of a graph G of order n with maximum degree ∆ are proved. Let Bn = K2 + Kn denote a book, i.e., the graph Bn consists of n triangles sharing an edge. The results are the following: (1) Let 1 < k ≤ l < ∆ < n and G be a connected {Bk+1,K2,l+1}-free graph of order n with maximum degree ∆. Then q(G) ≤ 1 4 [ 3∆ + k − 2l + 1...

Mantel’s theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let K− 4 = {123, 124, 134}, F6 = {123, 124, 345, 156} and F = {K− 4 , F6}: for n 6= 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tripartite 3-graph S3(n) (for n ...