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An independent set is a set of vertices in a graph, no two of which are adjacent. A maximum independent set is an independent set of the largest possible size for a given graph. The size of a maximum …

Graphs with minimum degree three that any two cycles have common vertex, have been characterized by Lovász. I see this result from the Plumer article （On the cyclic connectivity of planar graphs (197 …

Let an $i$-vertex be a vertex of degree $i$. Let an $i, j-$ edge be an edge joining an $i-$vertex to a $j-$vertex. Given a plane graph G, let $e_{i,j}$ be the number of $i, j-$edges of $G$. I found Bo …

I would like to construct a class of $k$-connected $k$-regular bipartite graphs with the girth at most $k-1$. This problem arises from a cycle. Any 2-connected 2-regular graph is a cycle, but its gi …

A signed graph is a graph in which each edge has a plus or minus sign. More specifically, a signed graph is a couple $S=(G, s)$, where $G=(V, E)$ is a graph with vertex set $V$ and edge set $E$, and $ …

A caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. From Wikipedia, I see that their count is also available in OEIS A005418. My question is, …

The line graph of an undirected graph $G$ is another graph $L(G)$ that represents the adjacencies between edges of $G$. $L(G)$ is constructed in the following way: for each edge in $G$, make a vertex …

The well-known max cut problem asks for a largest cut of a graph $G$. A cut of maximal size clearly corresponds to a bipartite subgraph of maximal size. After my inquiry, in planar graphs, the maximum …

I saw a question on the nauty emailing list without receiving any response, and it's something I've encountered in my own research as well. I am currently interested in graphs with diameter 3. I w …

For $k=2,3,4$, we solved this question. But for large $k$, we may need more deep tools. More details can be seen in L.C.Zhang, Y.Q. Huang, On the sizes of generalized cactus graphs, Discrete Applied …

Sometimes I look at all non-isomorphic good drawings of graphs on a plane or sphere. Good drawing means that no edge crosses itself, no two edges cross more than once, and no two edges incident with t …

Thanks for advice from Timothy Chow. I have now received an email from CRWS. The graph has a crossing number of 12. Its crossing-minimal drawing is as follows.

My question was partly inspired by the question linked below. There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....? I see a wonderful construction of Adam P. Goucher …

In graph theory, a cactus is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple …

From the Wikipedia entry on Tutte 12-cage , it is stated that the crossing number of Tutte 12-cage is 170, but the cited references do not seem to provide sufficient explanation for this. Exoo, G. " …