Related papers: 2-nearly Platonic graphs are unique

A nonplanar graph G is called almost-planar if for every edge e of G, at least one of G\e and G/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small…

A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs,…

The existence of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph $G$ is established, (i.e. each isomorphism between two copies of $K_4$ or $K_{2,2,2}$ in $G$ extends to an automorphism of $G$), with the 42 ordered lines of…

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few…

The Fiedler value $\lambda_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $\lambda_{2\max}$, and…

The deck of a graph $X$, $D(X)$, is defined as the multiset of all vertex-deleted subgraphs of $X$. Two graphs are said to be hypomorphic, if they have the same deck. Kelly-Ulam conjecture states that any two hypomorphic graphs on at least…

Distance-regular graphs are a class of regualr graphs with pretty combinatorial symmetry. In 2007, Miklavi\v{c} and Poto\v{c}nik proposed the problem of charaterizing distance-regular Cayley graphs, which can be viewed as a natural…

A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…

Let k be an algebraically closed field of characteristic p different from 2, and let F be a nodal surface of degree d in the projective 3-space P over k (i.e. the singularities of F are only ordinary quadratic, nodes for short). Let N be a…

We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the…

A connected graph $\G$ is said to be {\it distance-balanced} whenever for any pair of adjacent vertices $u,v$ of $\G$ the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$. In…

Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer…

A $k$-inner planar graph is a planar graph that has a plane drawing with at most $k$ {internal vertices}, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a $0$-inner planar graph. In…

Let G be a simple balanced bipartite graph on $2n$ vertices, $\delta = \delta(G)/n$, and $\rho={\delta + \sqrt{2 \delta -1} \over 2}$. If $\delta > 1/2$ then it has a $\lfloor \rho n \rfloor$-regular spanning subgraph. The statement is…

We show that every planar, 4-connected, K2;5-minor- free graph is the square of a cycle of even length at least six.

A matchstick graph is a graph drawn with straight edges in the plane such that the edges have unit length, and non-adjacent edges do not intersect. We call a matchstick graph $(m;n)$-regular if every vertex has only degree $m$ or $n$. In…

In this note, we prove that every 4-connected optimal 2-planar graph is Hamiltonian-connected. Furthermore, we show that the 4-connectedness condition is sharp by constructing infinitely many 3-connected optimal 2-planar graphs that are…

In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the…

For $S \subset \mathbb{R}^n$ and $d > 0$, denote by $G(S, d)$ the graph with vertex set $S$ with any two vertices being adjacent if and only if they are at a Euclidean distance $d$ apart. Deem such a graph to be ``non-trivial" if $d$ is…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…