Let S be a decomposition of a simple 4-regular plane graph into edge-disjoint cycles such that every two adjacent edges on a face belong to different cycles of S. Such graphs, called Grtzsch-Sachs graphs, may be considered as a result of a superposition of simple closed curves in the plane with tangencies disallowed.Koester studied the coloring of Grtzsch-Sachs graphs when all curves are circles.In 1984, he presented the first example of a 4-chromatic edge critical plane graph of order 40 formed by 7 circles.In the present paper, a new 4-chromatic edge critical graph generated by circles in the plane is presented.

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper we establish that X'irr(G)<= 4 for all colorable cactus graphs.

Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph $G$ is a mapping $\varphi : V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that the inequality $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $\varphi$ with $\max\{\varphi(v) : v \in V(G)\}$ = $k$. Das et al. [Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower bound for the radio number of graphs. In [Algorithms and Discrete Applied Mathematics: CALDAM 2019, Lecture Notes in Computer Science $\mathbf{11394}$, springer, Cham, 2019, 161-173], Bantva modified this technique for finding an improved lower bound on the radio number of graphs and gave a necessary and sufficient condition to achieve the improved lower bound. In this paper, one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graphs is given. Using this result, the radio number of the Cartesian product of a path and a wheel graphs is determined.

The suspension of the path $P_4$ consists of a $P_4$ and an additional vertex connected to each of the four vertices, and is denoted by $\hat{P_4}$. The largest number of triangles in a $\hat{P_4}$-free $n$-vertex graph is denoted by $ex(n,K_3,\hat{P_4})$. Mubayi and Mukherjee in 2020 showed that $ ex(n,K_3,\hat{P_4})= n^2/8+O(n)$. We show that for sufficiently large $n$, $ex(n,K_3,\hat{P_4})=\lfloor n^2/8\rfloor$.

We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern $\pi$.

The middle graph M (G) of a graph G is the graph obtained by subdividing each edge of G exactly once and joining all these newly introduced vertices of adjacent edges of G. A perfect Roman dominating function on a graph G is a function f : V (G) {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 is adjacent to exactly one vertex u for which f (u) = 2. The weight of a perfect Roman dominating function f is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on G. In this paper, a characterization of middle graphs with equal Roman domination and perfect Roman domination numbers is given.

A red-white coloring of a nontrivial connected graph G of diameter d is an assignment of red and white colors to the vertices of G where at least one vertex is colored red. Associated with each vertex v of G is a d-vector, called the code of v, whose ith coordinate is the number of red vertices at distance i from v. A red-white coloring of G for which distinct vertices have distinct codes is called an identification coloring or ID-coloring of G. A graph G possessing an ID-coloring is an ID-graph. The minimum number of red vertices among all ID-colorings of an ID-graph G is the identification number or ID-number of G. Necessary conditions are established for those trees that are ID-graphs. A tree T is starlike if T is obtained by subdividing the edges of a star of order 4 or more. It is shown that for every positive integer r different from 2, there exist starlike trees satisfying some prescribed properties having ID-number r.

Given a proper k-coloring of a graph G, a vertex v is locally recolorable if there is a proper k-coloring of the graph that changes the color of v and limits any other color changes to the neighbors of v. The coloring is robust if every vertex is locally recolorable. The robust chromatic number of G, R(G), is the smallest number k for which G has a robust k-coloring. If R(G) = (G), the graph is -robust and if deleting any vertex of a -robust graph decreases R(G), the graph is -robustcritical. We conjecture that only complete graphs are -robust-critical. This paper investigates this conjecture for = 4 and supports the conjecture for a large class of such graphs. Furthermore, conditions that must be satisfied for such graphs are determined.

We consider a variety of types of vertex sequences, which are defined in terms of a requirement that the next vertex in the sequence must meet. For example, let S = (v1, v2, . . . , v k ) be a sequence of distinct vertices in a graph G such that every vertex vi in S dominates at least one vertex in V that is not dominated by any of the vertices preceding it in the sequence S. Such a sequence of maximal length is called a dominating sequence since the set {v1, v2, . . . , v k } must be a dominating set of G. In this paper we survey the literature on dominating and other related sequences, and propose for future study several new types of vertex sequences, which suggest the beginning of a theory of vertex sequences in graphs.

Graph labeling is one of the most popular and dynamic areas of graph theory, perhaps even among all of mathematics. The standard problem involves a graph having labels from a given set of integers on its vertices and then are assigned values according to some formula. Frank Harary introduced an alternative problem, in which the labels on the vertices themselves must meet a specified condition. Here, we give a survey of a problem of each type: (a) strongly multiplicative graphs, where, given a labeling of the vertices, each edge is labeled with the product of its vertex labels; (b) sum graphs, where, given a labeling of the vertices, two are adjacent if the sum of their labels is also a vertex label.