A graph G is locally isometric if the subgraph induced by the neighbourhood of every vertex is an isometric subgraph of G. It is shown that the hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is NP-complete. Structural characterizations of locally isometric graphs, with maximum degree at most 6, that are fully cycle extendable, are established and these results are used to show that locally isometric graphs with maximum degree at most 6 are weakly pancyclic. This proves Ryjacek's conjecture for a subclass of locally connected graphs.
In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by Ostrovskii and Rosenthal and by Khukhro on minor exclusion for groups.
We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.
Magyar has shown that if $B \subset \mathbb{Z}^d$ has positive upper density $(d \geq 5)$, then the set of squared distances $\{ \|b_1-b_2 \|^2 \text{ }: \text{ } b_1,b_2 \in B \}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements …
We prove that for any given $\varepsilon>0$ and $d\in [0,1]$, every sufficiently large $(\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being $(\varepsilon, d)$-dense means that every set $X$ containing at least~$\varepsilon\,|V(G)|$ vertices spans at least $\tfrac d2\, |X|^2$ edges, and what we really count is the number of homomorphisms from an $r$-cycle into $G$. The result adresses a question of Y. Kohayakawa, B. Nagle, V. Rödl, and M. Schacht.
Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube $G$ with girth more than 6 is a tree-zone graph and therefore $2n(G)-m(G)-i(G)+ce(G)=2$ holds, where $i(G)$ is the isometric dimension of $G$ and $ce(G)$ its convex excess.
A graph is $1$-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal $1$-planar. For a $3$-connected locally maximal $1$-planar graph $G$, we show the existence of a spanning $3$-connected planar subgraph and prove that $G$ is hamiltonian if $G$ has at most three $3$-vertex-cuts, and that $G$ is traceable if $G$ has at most four $3$-vertex-cuts. Moreover, infinitely …
The clustering coefficient of a vertex in a graph is the proportion of neighbours of the vertex that are adjacent. The minimum clustering coefficient of a graph is the smallest clustering coefficient taken over all vertices. A complete structural characterization of those locally connected graphs, with minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully cycle extendable is given in terms of strongly induced subgraphs with given attachment sets. Moreover, it is shown that all locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most …
The research in the present paper was motivated by the conjecture of Ryjáček that every locally connected graph is weakly pancyclic. For a connected locally connected graph $G$ of order at least $3$, our results are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is weakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycle extendable if and only if $2δ(G)\geq n(G)$. If $G$ is $\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$-free or $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$-free, then $G$ is fully cycle extendable. If $G$ is distinct from $K_1+K_1+\bar{K}_3$ and $\{ K_1+P_4,K_…
The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of $\ell_1$ spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal …
A graph $Γ$ is $k$-connected-homogeneous ($k$-CH) if $k$ is a positive integer and any isomorphism between connected induced subgraphs of order at most $k$ extends to an automorphism of $Γ$, and connected-homogeneous (CH) if this property holds for all $k$. Locally finite, locally connected graphs often fail to be 4-CH because of a combinatorial obstruction called the unique $x$ property; we prove that this property holds for locally strongly regular graphs under various purely combinatorial assumptions. We then classify the locally finite, locally connected 4-CH graphs. We also classify the l…
We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomassé.
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph c…
