In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.
In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We define the pegging number as the smallest number of pegs needed to reach all the vertices in a graph no matter what the distribution. Similarly, the optimal-pegging number of a graph is defined as the smallest distribution of pegs for which all the vertices in the graph can be reached. We obtain tight bounds on the pegging numbers and optimal-pegging numbers of …
In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph. Using a similar construction, we also prove the conjecture that the zero-forcing number of a graph is no more than the zero-forcing number of its line graph; moreover we prove that the brushing number of a graph is no more than the brushing number of its line graph. All three bounds are shown to be tight.
In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double start graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.
In this paper, we study the notions of connected domination number and of outer-connected domination number for middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(G). We also compute the outer-connected domination number of some families of graphs such as star graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and some operation on graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the outer-connected domination number of middle graphs.
The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number. We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These i…
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.
Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed independence number.
In the paper we give a lower bound for the number of vertices of a given graph using its chromatic number. We find the graphs for which this bound is exact. The results are applied in the theory of Foklman numbers.
We prove that the graphs $T\boxslash P$ have unbounded stack number and queue number $3$, where $T$ is a tree and $P$ is a path, and $\boxslash$ denotes the graph strong product but with one of the directions removed. The previous best known results is that graphs with queue number $4$ can have unbounded stack number.
The matching number of a graph G is the size of a maximum matching in the graph. In this note, we present a sufficient condition involving the matching number for the Hamiltonicity of graphs.
Computing the clique number and chromatic number of a general graph are well-known NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. \emph{Linear Algebra Appl.}, 285(1-3): 123--142, 1998) showed that computing clique number and chromatic number are still NP-Hard problems for the class of circulant graphs. We show that computing clique number is NP-Hard for the class of Cayley graphs for the groups $G^n$, where $G$ is any fixed finite group (e.g., cubelike graphs). We a…
Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
Let $r\geq 3$ be an integer and $G$ be a graph. Let $δ(G), Δ(G)$, $α(G)$ and $μ(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper proved $δ(G)α(G)\leq Δ(G)μ(G)$. Mohr and Rautenbach characterized the extremal graphs for non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs for all $r$-regular graphs in term of Gallai-Edmonds Structure Theorem, which extends Mohr and Rautenbach's result.
We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.
We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields 7 graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.
We investigate a generalization of the bondage number of a graph called the \textit{$k\,$-synchronous bondage number}. The $k\,$-synchronous bondage number of a graph is the smallest number of edges that, when removed, increases the dominating number by $k$. In this paper, we discuss the 2-synchronous bondage number and then generalize to $k\,$-synchronous bondage number. We present $k\,$-synchronous bondage number for several graph classes and give bounds for general graphs. We propose this characteristic as a metric of the connectivity of a simple graph with possible uses in the field of net…
The exact crossing number is only known for a small number of families of graphs. Many of the families for which crossing numbers have been determined correspond to cartesian products of two graphs. Here, the cartesian product of the Sunlet graph, denoted $\mathcal{S}_n$, and the Star graph, denoted $K_{1,m}$, is considered for the first time. It is proved that the crossing number of $\mathcal{S}_n \Box K_{1,2}$ is $n$, and the crossing number of $\mathcal{S}_n \Box K_{1,3}$ is $3n$. An upper bound for the crossing number of $\mathcal{S}_n \Box K_{1,m}$ is also given.
The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the bounds on the sum and product of the independence numbers of graphs and their line graphs.
A graph G on n vertices is said to be extendable if G can be modified to form a new graph H on more than n vertices, while preserving the degrees of the vertices common to G and H. The added vertices all have the same degree and we define edit numbers to quantify the amount of modification needed to obtain the extended graph. Characterizing graphs with least possible edit numbers, we obtain that graphs with zero edit number can be extended using regular graphs. We also describe an iterative algorithm to construct connected regular graphs on arbitrarily large vertex sets, starting from the comp…
We introduce the triple crossing number, a variation of crossing number, of a graph, which is the minimal number of crossing points in all drawings with only triple crossings of the graph. It is defined to be zero for a planar graph, and to be infinite unless a graph admits a drawing with only triple crossings. In this paper, we determine the triple crossing numbers for all complete multipartite graphs including all complete graphs.
A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.
In an earlier paper, the present authors (2013) introduced the altermatic number of graphs and used Tucker's Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the altermatic number is a lower bound for the chromatic number. A matching graph has the set of all matchings of a specified size of a graph as vertex set and two vertices are adjacent if the corresponding matchings are edge-disjoint. It is known that the Kneser graphs, the Schrijver graphs, and the permutation graphs can be represented by matching graphs. In this paper, as a generalization of the well-…
We explore various techniques for counting the number of straight-edge crossing-free graphs that can be embedded on a planar point set. In particular, we derive a lower bound on the ratio of the number of such graphs with $m+1$ edges to the number of graphs with $m$ edges. We show how a relatively small improvement of this bound would improve existing bounds concerning numbers of plane graphs with a prescribed number of edges. Furthermore, we use a cross-graph charging scheme to derive lower bounds for the number of such graphs when the point set has few extreme points.
For a given graph $F$, the $F$-saturation number of a graph $G$ is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. Recently, the $F$-saturation number of the Erdős$\text{\bf--}$Rényi random graph $\mathbbmsl{G}(n, p)$ has been determined asymptotically for any complete graph $F$. In this paper, we give an asymptotic formula for the $F$-saturation number of $\mathbbmsl{G}(n, p)$ when $F$ is a star graph.
The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for $\text{sg}(K_{n,m})$ is given, as well as a formula for the crown graphs $S_n^0$. Bounds on $\text{sg}(Q_n)$ are also discussed.
We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such that each ball of radius $r$ is properly $c$-colorable, we have $χ(G)\leq n$.
We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given order, size and number of connected components.
The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced forests of $G$. The line graphs of trees are the claw-free block graphs. These graphs have been used by Erdős, Saks and Sós to construct graphs with a given number of edges and vertices whose maximum induced tree is very small. In this paper, we give bounds on the decycling number of line graphs of trees and construct extremal trees to show that these bounds are …
In this short note, we give a description of the Parry-Sullivan number of a graph in terms of the cycles in the graph. This tool is occasionally useful in reasoning about the Parry-Sullivan numbers of graphs.
We show that the independence number of a countably infinite HH-homogeneous graph that does not contain the Rado graph as a spanning subgraph is finite and present a classification of MB-homogeneous graphs up to bimorphism-equivalence as a consequence.
We prove that every triangle-free graph with maximum degree $Δ$ has list chromatic number at most $(1+o(1))\fracΔ{\ln Δ}$. This matches the best-known bound for graphs of girth at least 5. We also provide a new proof that for any $r\geq 4$ every $K_r$-free graph has list-chromatic number at most $200r\frac{Δ\ln\lnΔ}{\lnΔ}$.
Bubble-sort star graphs are a combination of star graphs and bubble sort graphs. They are bipartite graphs and also form a family of Cayley graphs. The decycling number of a graph is the minimum number of vertices whose removal from the graph results in an acyclic subgraph. In this paper, we prove the decycling number D(n) of an n-dimensional bubble-sort star graph for n <= 5. We also show D(n) satisfies the inequalities for n >= 6.
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.
In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.
We highlight new results on the localization number of a graph, a parameter derived from the localization graph searching game. After introducing the game and providing an overview of existing results, we describe recent results on the localization number. We describe bounds or exact values of the localization number of incidence graphs of designs, polarity graphs, and Kneser graphs.
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 König-Egerváry graphs in terms of their matching numbers.
The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.
The "minor crossing number" of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.
The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimal number of edges not in $M$ whose removal to make $M$ as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect matchings of $G$ is the anti-forcing spectrum of $G$. In this paper, we characterize the plane elementary bipartite graph whose minimum anti-forcing number is one. We show that the maximum anti-forcing number of a graph is at most its cyclomatic number. In particular, we characterize the graphs with the maximum anti-forcing number achieving the upper bound, such ex…
It is shown that for $n\geq 141$, among all triangle-free graphs on $n$ vertices, the complete equibipartite graph is the unique triangle-free graph with the greatest number of cycles.
Let $K_{r_1,\ldots,r_s}$ denote the complete multipartite graph with class sizes $r_1,\ldots,r_s$ and let $K_s$ denote the complete graph of order $s$. In 2018, Luo determined the maximum number of $K_s$ in 2-connected graphs with a given circumference. Recently, Lu, Yuan and Zhang determined the maximum number of $K_{r_1,r_2}$ in 2-connected graphs with a given circumference, and Wang determined the maximum number of $K_s$ or $K_{r_1,1,\ldots,1}$ in graphs with given matching number. Motivated by these works, we determine the maximum number of $K_{r_1,\ldots,r_s}$ in $2$-connected graphs with…
The structure of all triangle free graphs G = (V,E) with |E| - 6|V| + 13α(G) = 0 is determined, yielding an affirmative answer to a question of Stanisław Radziszowski and Donald Kreher.
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with one same color. An edge-colored graph is called $k$-proper connected if any two vertices of the graph are connected by $k$ internally pairwise vertex-disjoint proper paths in the graph. The $k$-proper connection number of a $k$-connected graph $G$, denoted by $pc_k(G)$, is defined as the smallest number of colors that are needed in order to make $G$ $k$-proper connected. For $k=1$, we write $pc(G)$ other than $pc_1(G)$, and call it the proper connection number of $G$. In this paper, we…
We introduce joins of signed graphs and explore the chromatic number of the all-positive and all-negative joins. We prove an analogue to the theorem that the chromatic number of the join of two graphs equals the sum of their chromatic numbers. Given two signed graphs, the chromatic number of the all-positive and all-negative join is usually less than the sum of their chromatic numbers, by an amount that depends on the new concept of deficiency of a signed-graph coloration.
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number of components is restricted and illustrates a connection between graph compositions and Stirling numbers of the second kind.
A total dominator coloring of a graph G is a proper coloring of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this article, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitely the total dominator chromatic number of the middle graph of several known families of graphs.
