For the class of triangle-free graphs Brooks’ Theorem can be restated in terms of forbidden induced subgraphs, i.e. let G be a triangle-free and K1,r+1-free graph. Then G is r-colourable unless G is isomorphic to an odd cycle or a complete graph with at most two vertices. In this note we present an improvement of Brooks’ Theorem for triangle-free and rsunshade-free graphs. Here, an r-sunshade (...

An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every n-vertex graph has at most 10 ≈ 1.5849 maximal induced matchings, and this bound is best possible. We prove that every n-vertex triangle-free graph has at most 3 ≈ 1.4423 maximal induced matchings, and this bound is attained by every disjoint union of copies of the complete bipar...

In a graph G, the first and second degrees of a vertex v is equal to thenumber of their first and second neighbors and are denoted by d(v/G) andd 2 (v/G), respectively. The first, second and third leap Zagreb indices are thesum of squares of second degrees of vertices of G, the sum of products of second degrees of pairs of adjacent vertices in G and the sum of products of firs...

A graph G is pancyclic if it contains cycles of each length `, 3 ≤ ` ≤ |V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths Pi+1 and Pj+1 with distinct vertices of a triangle. Gould, Luczak and Pfender [4] showed that if G is a 3-connected {K1,3, B(i, j)}-free graph with i + j = 4 then G is pancyclic. In this paper, we prove that every 4-connected, ...

It is proven that if G is a 3-connected claw-free graph which is also Z3-free (where Z3 is a triangle with a path of length 3 attached), P6-free (where P6 is a path with 6 vertices) or H1-free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L such that if a 3-connected ...

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai [Z.-H. Chen, H.-J. Lai, Reduction techniques for super-Eulerian graphs and related topics— an update, in: Ku Tung-Hsin (Ed.), Combinatorics and Graph Theory, vol. 95, World Scientific, Singapore/London, 1995, pp. 53–69, Conjecture 8.6] conjectured that every 3-edge connected, essentially 6-edge connected graph ...

Given two graphs H1 and H2, a graph G is (H1,H2)-free if it contains no subgraph isomorphic to H1 or H2. Let Pt and Cs be the path on t vertices and the cycle on s vertices, respectively. In this paper we show that for any (P6, C4)-free graph G it holds that χ(G) ≤ 3 2 ω(G), where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. Our bound is attained by several graph...

Let G be a 2-connected claw-free graph. We show that • if G is N1,1,4-free or N1,2,2-free or Z5-free or P8-free, respectively, then G has a spanning eulerian subgraph (i.e. a spanning connected even subgraph) or its closure is the line graph of a graph in a family of well-defined graphs, • if the minimum degree δ(G) ≥ 3 and G is N2,2,5-free or Z9-free, respectively, then G has a spanning euleri...

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order ν and minimum degree δ ≥ ν+6 10 is Hamiltonian for ν sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order ν, and if δ(H) ≥ ν+5 10 , then either H is hamiltonian, or δ(H) = ν+5 10 and the Ryjác̆ek’s closure cl(H) of H is the line graph of a graph obtained...