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.     

List coloring triangle-free planar graphs

This paper proves the following result. Assume $G$ is a triangle-free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\text{◂=▸}|L\left(v\right)|=4$ for each vertex $\text{◂+▸}v\in V\left(G\right)-X$ and $\text{◂=▸}|L\left(v\right)|=3$ for each vertex $v\in X$, then $G$ is $L$-colorable.     

Extending regular edge-colorings of complete hypergraphs

A coloring (partition) of the collection $(\genfrac{}{}{0.0pt}{}{X}{h})$ of all $h$-subsets of a set $X$ is $r$-regular if the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. We are interested in finding the conditions under which a given $r$-regular coloring of $(\genfrac{}{}{0.0pt}{}{X}{h})$ is extendible to an $s$-regular coloring of $(\genfrac{}{}{0.0pt}{}{Y}{h})$ for $s⩾r$ and $Y⊋X$. The case $\text{◂,▸}h=2,r=s=1$ was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for $h⩾3$. The case $r=s=1$ was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases $h=2$ and $h=3$ were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases $h=4,\text{◂⩾▸}|Y|⩾4|X|$ and $h=5,\text{◂⩾▸}|Y|⩾5|X|$.     

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.     

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.     

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 $\mathrm{\Delta }$.