Computability of graphs

We consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type, which means that they have the following property: if X is a computable topological space and $f:A\to X$ a topological imbedding such that $f(A)$ and $f(B)$ are semicomputable sets in X, then $f(A)$ is a computable set in X. It is known, e.g., that $(M,\partial M)$ has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that $(G,E)$ has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.     

Strong cliques in vertex-transitive graphs

A clique (resp, independent set) in a graph is strong if it intersects every maximal independent set (resp, every maximal clique). A graph is clique intersect stable set (CIS) if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique $C$ in a vertex-transitive graph $\mathrm{\Gamma }$ is strong if and only if $\text{◂=▸}\text{◂⋅▸}\mid C\mid \mid I\mid =\mid V\left(\mathrm{\Gamma }\right)\mid$ for every maximal independent set $I$ of $\mathrm{\Gamma }$. On the basis of this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transitive graphs of valency at most 4 admitting a strong clique, and give a partial characterization of 5-valent vertex-transitive graphs admitting a strong clique. Our results imply that every vertex-transitive graph of valency at most 5 that admits a strong clique is localizable. We answer an open question by providing an example of a vertex-transitive CIS graph which is not localizable.     

A characterization of (4,2)-choosable graphs

A graph $G$ is $\left(a,b\right)$-choosable if given any list assignment $L$ with $\text{◂=▸}\mid L\left(v\right)\mid =a$ for each $\text{◂+▸}v\in V\left(G\right)$ there exists a function $\phi$ such that $\text{◂⊆▸}\phi \left(v\right)\subseteq L\left(v\right)$ and $\text{◂=▸}\mid \phi \left(v\right)\mid =b$ for all $\text{◂+▸}v\in V\left(G\right)$, and whenever vertices $x$ and $y$ are adjacent $\text{◂+▸}\phi \left(x\right)\cap \phi \left(y\right)=\varnothing$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.     

The optimal general upper bound for the distinguishing index of infinite graphs

The distinguishing index $D\prime \left(G\right)$ of a graph $G$ is the least cardinal number $d$ such that $G$ has an edge-coloring with $d$ colors, which is preserved only by the trivial automorphism. We prove a general upper bound $D\prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }-1$ for any connected infinite graph $G$ with finite maximum degree $\mathrm{\Delta }\ge 3$. This is in contrast with finite graphs since for every $\mathrm{\Delta }\ge 3$ there exist infinitely many connected, finite graphs $G$ with $\text{◂,▸}D\prime \left(G\right)=\mathrm{\Delta }$. We also give examples showing that this bound is sharp for any maximum degree $\mathrm{\Delta }$.     

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.