Related papers: Graphs with girth at least 5 with orders between 2…
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…
The "slope-number" of a graph $G$ is the minimum number of distinct edge slopes in a straight-line drawing of $G$ in the plane. We prove that for $\Delta\geq5$ and all large $n$, there is a $\Delta$-regular $n$-vertex graph with…
The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$…
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of sets such that $|L(v)|\ge 5$ for every $v\in V(G)$. By an $L$-coloring of a subgraph $J$ of $G$ we mean a (proper) coloring $\phi$ of $J$ such that…
We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.
In 2010 it was proved that a 3-regular matchstick graph of girth 5 must consist at least of 30 vertices. The smallest known example consisted of 180 vertices. In this article we construct an example consisting of 54 vertices and prove its…
We prove a conjecture of Dvo\v{r}\'ak, Kr\'al, Nejedl\'y, and \v{S}krekovski that planar graphs of girth at least five are square $(\Delta+2)$-colorable for large enough $\Delta$. In fact, we prove the stronger statement that such graphs…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…
Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] determined the unique spectral extremal graph over all outerplanar graphs and the unique spectral extremal graph over all planar graphs when the number of vertices is…
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…
For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a…
We give an upper bound for the online chromatic number of graphs with high girth and for graphs with high oddgirth generalizing Kier- stead's algorithm for graphs that contain neither a C3 or C5 as an induced subgraph.
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…
An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a…
Let $v(F)$ denote the number of vertices in a fixed connected pattern graph $F$. We show an infinite family of patterns $F$ such that the existence of a subgraph isomorphic to $F$ is expressible by a first-order sentence of quantifier depth…
The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…
We investigate prime character degree graphs of solvable groups that have six vertices. There are one hundred twelve non-isomorphic connected graphs with six vertices, of which all except nine are classified in this paper. We also…