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Register a new userEvery finite group has a normal bi-Cayley graph
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Jin-Xin Zhou
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A graph $G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $Aut(G)$, we say that $G$ is {em normal} with respect to $H$. In this paper, we show that every finite group has a connected normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri, Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides a positive answer to the Question of the above paper.
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A graph $G$ admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {em bi-Cayley graph/} over $H$. Such a graph $G$ is called {em normal/} if $H$ is normal in the full automorphism group of $G$, and {em normal edge-transitive/} if the normaliser of $H$ in the full automorphism group of $G$ is transitive on the edges of $G$. % In this paper, we give a characterisation of normal edge-transitive bi-Cayley graphs, %which form an important subfamily of bi-Cayley graphs, and in particular, we give a detailed description of $2$-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic $p$-groups. We find that under certain conditions, `normal edge-transitive is the same as `normal for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most $6$, and answer some open questions from the literature about $2$-arc-transitive, half-arc-transitive and semisymmetric graphs.
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.
We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $mathbb{Z}_2^d$ and $mathbb{Z}_4^d$. We also provide some infinite families of Cayley graphs of $mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
A graph $G=(V,E)$ is total weight $(k,k)$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k$ real numbers, there is a proper $L$-total weighting, i.e., a map $phi: V cup E to mathbb{R}$ such that $phi(z) in L(z)$ for $z in V cup E$, and $sum_{e in E(u)}phi(e)+phi(u) e sum_{e in E(v)}phi(e)+phi(v)$ for every edge ${u,v}$. A graph is called nice if it contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture, it was conjectured in [T. Wong and X. Zhu, Total weigt choosability of graphs, J. Graph Th. 66 (2011),198-212] that every nice graph is total weight $(1,3)$-choosable. The problem whether there is a constant $k$ such that every nice graph is total weight $(1,k)$-choosable remained open for a decade and was recently solved by Cao [L. Cao, Total weight choosability of graphs: Towards the 1-2-3 conjecture, J. Combin. Th. B, 149(2021), 109-146], who proved that every nice graph is total weight $(1, 17)$-choosable. This paper improves this result and proves that every nice graph is total weight $(1, 5)$-choosable.
We prove that every tetrahedron T has a simple, closed quasigeodesic that passes through three vertices of T. Equivalently, every T has a face whose exterior angles are at most pi.
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