Graphs without cycles of even length

On list vertex 2-arboricity of toroidal graphs without cycles of specific length

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

Even cycles in graphs

A Note on Graphs Without Short Even Cycles

In this note, we show that any n-vertex graph without even cycles of length at most 2k has at most 12n 1+1/k + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k ∈ {2, 3, 5}.

Even Cycles in Directed Graphs

It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)/4⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.

On relating edges in graphs without cycles of length 4

An edge xy is relating in the graph G if there is an independent set S, containing neither x nor y, such that S ∪ {x} and S ∪ {y} are both maximal independent sets in G. It is an NP-complete problem to decide whether an edge is relating [1]. We show that the problem remains NP-complete even for graphs without cycles of length 4 and 5. On the other hand, for graphs without cycles of length 4 and...