Related papers: A Classification of Orientable Regular Embeddings …
The regular embeddings of complete bipartite graphs $K_{n,n}$ in orientable surfaces are classified and enumerated, and their automorphism groups and combinatorial properties are determined. The method depends on earlier classifications in…
An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge…
A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the…
In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets of complete bipartite graphs $K_{m,n}$ by solving the cases where $n=1$, $2$, $3$, $4$, and $5$, and also the case where $m=n$. In an article…
In 2009, Kong, Wang, and Lee introduced the problem of finding the edge-balanced index sets ($EBI$) of complete bipartite graphs $K_{m,n}$, where they examined the cases $n=1$, $2$, $3$, $4$, $5$ and the case $m=n$. Since then the problem…
The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and…
In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets (EBI) of complete bipartite graphs K_{m,n} by solving the cases where n = 1, 2, 3, 4, and 5, and also the case where m = n. In 2011, Krop and…
A complete bipartite graph $K_{3,3}$, considered as a planar linkage with joints at the vertices and with rods as edges, in general admits only motions as a whole, i.e., is inflexible. Two types of its paradoxical mobility were found by…
A graph $G$ is said to be $1$-perfectly orientable if it has an orientation such that for every vertex $v\in V(G)$, the out-neighborhood of $v$ in $D$ is a clique in $G$. In $1982$, Skrien posed the problem of characterizing the class of…
The study of ORR was inspired by L\'{a}zsl\'{o} Babai in 1980 when he asked a question: Which [finite] groups admit an oriented graph as a DRR? And it has been solved by Joy Morris and Pablo Spiga through a series of papers in 2018. In this…
An oriented graph is an orientation of a simple graph. In 2009, Keevash, K\"{u}hn and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $(3n-4)/8$ is Hamiltonian. Later, Kelly, K\"{u}hn and Osthus showed that…
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc…
In a simple drawing of a graph every pair of edges intersect each other in at most one point, which is either a common endvertex or a proper crossing. For each positive integer $n$, Negami identified a drawing $B_n$ of the complete…
Alspach [{\sl Bull. Inst. Combin. Appl.}~{\bf 52} (2008), 7--20] defined the maximal matching sequencibility of a graph $G$, denoted~$ms(G)$, to be the largest integer $s$ for which there is an ordering of the edges of $G$ such that every…
Mark Jungerman's 1975 Ph.D. thesis presents several infinite families of index 2 current graphs that generate triangular embeddings of complete or near-complete graphs. However, there is one family mentioned, the complete graphs on $36s$…
Let $G$ be a graph of order $n$ and let $k\in \{1,2,\ldots,n-1\}$. The $k$-token graph of $G$ is the graph, whose vertices are all the $k$-subsets of vertices of $G$, where two such $k$-sets are adjacent whenever their symmetric difference…
This note gives a detailed proof of the following statement. Let $d\in \mathbb{N}$ and $m,n \ge d + 1$, with $m + n \ge \binom{d+2}{2} + 1$. Then the complete bipartite graph $K_{m,n}$ is generically globally rigid in dimension $d$.
Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the…
We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n,m}$ can be induced by a homeomorphism of some embedding of the graph in $S^3$.
Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…