Family Of Finite Graphs Research Articles

On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs F, a graph A and a positive integer r, if every graph B∈F has a 2-vertex-connected subgraph which is not a subgraph of A, then there exists an F-free graph which is vertex r-Ramsey with respect to A. We prove that this sufficient condition for the existence of an F-free graph which is vertex r-Ramsey with respect to A is also necessary for large enough number of colours r.We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.

Linear Kernels for Edge Deletion Problems to Immersion-Closed Graph Classes

Suppose ${\mathcal{F}}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion...

The typical structure of graphs without given excluded subgraphs

AbstractLet $ {\cal L}$ be a finite family of graphs. We describe the typical structure of $ {\cal L}$‐free graphs, improving our earlier results (Balogh et al., J Combinat Theory Ser B 91 (2004), 1–24) on the Erdős–Frankl–Rödl theorem (Erdős et al., Graphs Combinat 2 (1986), 113–121), by proving our earlier conjecture that, for $ p = p ({\cal L}) = {\rm min}_L \in {\cal L} \chi (L) - 1 $, the structure of almost all $ {\cal L}$‐free graphs is very similar to that of a random subgraph of the Turán graph Tn,p. The “similarity” is measured in terms of graph theoretical parameters of $ {\cal L}$.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

New Lower Bound for Multicolor Ramsey Numbers for Even Cycles

For given finite family of graphs $G_{1}, G_{2}, \ldots , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, \ldots , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors then there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. We give a lower bound for $k-$color Ramsey number $R(C_{m}, C_{m}, \ldots , C_{m})$, where $m \geq 4$ is even and $C_{m}$ is the cycle on $m$ vertices.

A weak immersion relation on graphs and its applications

Weak immersion is a generalization of the immersion relation defined by Nash-Williams. A graph H is said to be weakly immersed in a graph G if H can be obtained from G by a sequence of these three operations: taking a subgraph, splitting a vertex, and lifting a pair of adjacent edges. The weak immersion relation has the useful property that finite graphs are well-quasi-ordered by it, which also holds for graphs with some vertices designated as terminals. As a result, any family of finite graphs that is closed under weak immersion can be characterized by a finite number of minimal forbidden graphs called obstructions. Weak immersion offers two advantages over immersion for practical applications. First, although closure under weak immersion implies closure under immersion, families can have significantly fewer obstructions under weak immersion. Hence weak immersion can provide simpler characterizations for closed families. Examples include graphs of bounded cutwidth and graphs of bounded multiway cutsize. The difference in the number of obstructions is at least exponential in the cutwidth and in the square-root of the multiway cutsize. Second, for every fixed graph H, there is a polynomial-time algorithm to decide whether H is weakly immersed in an input graph G. Consequently, there is a polynomial-time membership test for any family that is closed under weak immersion. In principle, testing for weak immersion is as fast as testing for immersion. Thus the simpler characterization provided by weak immersion may lead to faster membership algorithms.

Counterexamples to two conjectures about distance sequences

It is shown that, contrary to a pair of well-known conjectures, there exist finite and infinite examples of: (1) vertex-transitive graphs whose distance sequences are not unimodal, and (2) graphs with primitive automorphism group whose distance sequences are not logarithmically convex. In particular, a family of finite graphs is presented whose automorphism groups are primitive and whose distance sequences are not unimodal.