Graph homomorphism reconfiguration and frozen H-colorings

We say that a graph $F$ strongly arrows a pair of graphs $\left(G,H\right)$ and write $\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)$ if any coloring of its edges with red and blue leads to either a red $G$ or a blue $H$ appearing as induced subgraphs of $F$. The induced Ramsey number, $\text{◂...▸}\text{IR}\left(G,H\right)$, is defined as $\text{◂lim▸}\min \text{◂{}▸}\{|V\left(F\right)|:\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)\}$. We consider the connection between the induced Ramsey number for a pair of two connected graphs $\text{◂...▸}\text{IR}\left(G,H\right)$ and the induced Ramsey number for multiple copies of these graphs $\text{IR}\text{◂()▸}\left(sG,tH\right)$, where $xG$ denotes the pairwise vertex-disjoint union of $x$ copies of $G$. It is easy to see that if $\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)$, then $\text{◂⋅▸}\left(s+t-1\right)F\stackrel{\text{ind}}{\to }\text{◂()▸}\left(sG,tH\right)$. This implies that $$\text{◂...▸}\text{IR}\text{◂≤▸}\text{◂()▸}\left(sG,tH\right)\le \left(s+t-1\right)\text{IR}\left(G,H\right).$$ For several specific graphs, such as a path on three vertices vs a complete multipartite graph, a matching vs a complete graph, or a matching vs another matching, it is known that the above inequality is tight. On the other hand, the inequality is also strict for some graphs. However, even in the simplest case when $H$ is an edge and $t=2$, it is not known for what $G$ and for what $s$ the above inequality is tight. We show that it is tight if $G$ is connected and $s$ is at least as large as the order of $G$. In addition, we make further progress in determining induced Ramsey numbers for multiple copies of graphs, such as paths and triangles.     

Graph-like compacta: Characterizations and Eulerian loops

A compact graph-like space is a triple $\left(X,V,E\right)$, where $X$ is a compact, metrizable space, $V\subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X⧹V\cong \text{◂+▸}E\times \left(0,1\right)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.