In this paper, we study matching integral graphs of small order. A graph is called matching integral if the zeros of its matching polynomial are all integers. Matching integral graphs were first studied by Akbari, Khalashi, etc. They characterized all traceable graphs which are matching integral. They studied matching integral regular graphs. Furthermore, it has been shown that there is no matc...

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, [8] proved that planar isomorphism is complete for log-space. We extend this result further to the classes of graphs which exclude K3,3 or K5 as a minor, and give a log-space algorithm. Our algo...

Thomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and...

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] gave such a graph which is not 3-choosable. We prove that every triangle-free planar graph such that 4-cycles do not share edges with other 4and 5-cycles is 3-choosable. T...

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a speciic innnite sequence of 3-colourable graphs or it contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that if a (not necessarily maximal) triangle-free graph of order n has minimum degree n=...

The generalized bull is the graph B i , j ${B}_{i,j}$ obtained by attaching endvertices of two disjoint paths lengths $i,j$ to vertices a triangle. We prove that every 3-connected { K 1 3 X } $\{{K}_{1,3},X\}$ -free graph, where ? 6 2 5 4 $X\in \{{B}_{1,6},{B}_{2,5},{B}_{3,4}\}$ Hamilton-connected. results are sharp and complete characterization forbidden induced bulls implying Hamilton-connect...

We show that every 3-connected claw-free graph which contains no induced copy of P11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P12 this result is, in a way, best possible. 1. Statement of the main result A graph G is {H1, H2, . . . Hk}-free if G contains no induced subgraphs isomorphic to any of the graphs Hi, i = 1, 2, . . . , k. A...

A graph G is said to be pancyclic if G contains cycles of all lengths from 3 to |V (G)|. We show that if G is 4-connected, claw-free, and P10-free, then G is either pancyclic or it is the line graph of the Petersen graph. This implies that every 4-connected, claw-free, P9-free graph is pancyclic, which is best possible and extends a result of Gould et al. Pancyclicity in 3-connected graphs: Pai...