The spectral radius of graphs with no intersecting odd cycles

Extremal Graph for Intersecting Odd Cycles

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let Tn,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that Tn,r is the only extremal graph for complete graph Kr+1. Erdős et al. (1995) determined the ...

A spectral condition for odd cycles in graphs

Let G be a graph of su¢ ciently large order n; and let the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c: Then G contains a cycle of length t for every t n=320: This condition is sharp: the complete bipartite graph T2 (n) with parts of size bn=2c and dn=2e contains no odd cycles and its largest eigenvalue is equal to p bn2=4c: This condition is stable: if (G) is close t...

The distance spectral radius of graphs with given number of odd vertices

On the Odd Cycles of Normal Graphs

A graph is normal if there exists a cross-intersecting pair of set families one of which consists of cliques while the other one consists of stable sets, and furthermore every vertex is obtained as one of these intersections. It is known that perfect graphs are normal while Cs, CT. and (_: are not. We conjecture that these three graphs are the only minimally not normal graphs. We give sufficien...

K4-free graphs with no odd holes

All K4-free graphs with no odd hole and no odd antihole are three-colourable, but what about K4free graphs with no odd hole? They are not necessarily three-colourable, but we prove a conjecture of Ding that they are all four-colourable. This is a consequence of a decomposition theorem for such graphs; we prove that every such graph either has no odd antihole, or belongs to one of two explicitly...