Related papers: How Much Regularity Forces a Sequence to be Graphi…
Let $G=(V,E)$ be a graph of order $n$ and let $1\leq k< n$ be an integer. The $k$-token graph of $G$ is the graph whose vertices are all the $k$-subsets of $V$, two of which are adjacent whenever their symmetric difference is a pair of…
A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that $1$-planar graphs have at most $4n-8$ edges. We prove the following odd-even generalization. If a graph can be…
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…
Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which…
In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for…
The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…
For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges…
To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic…
A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $b(G)$ for which $G$ has a b-coloring…
A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and…
Let $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$,…
In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the…
In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are…
A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover…
The definition of edge-regularity in graphs is a relaxation of the definition of strong regularity, so strongly regular graphs are edge-regular and, not surprisingly, the family of edge-regular graphs is much larger and more diverse than…
A graph is called odd if all of its vertex degrees are odd. A long-standing conjecture asked whether there exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph on at…
Let $\pi_1=(d_1^{(1)}, \ldots,d_n^{(1)})$ and $\pi_2=(d_1^{(2)},\ldots,d_n^{(2)})$ be graphic sequences. We say they \emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $\pi_1$ and $\pi_2$, respectively, on vertex set…
We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used…