Recently, it was shown that the Gallai-Ramsey number satisfies \(gr(F_{3,2}, K_3, K_3)=31\), where \(F_{3,2}\) is the generalized fan \(F_{3,2}:=K_1+2K_3\). In this paper, we show that the star-critical Gallai-Ramsey number satisfies \(gr_*(F_{3,2}, K_3, K_3)=27\). We also prove that the critical colorings for \(r_*(K_3, K_3)\), \(gr(F_{3,2},K_3,K_3)\), and \(gr_*(F_{3,2},K_3,K_3)\) are unique.

Richard Rado’s work in Ramsey Theory established conditions under which monochromatic solutions to a linear system must occur. In this paper, we find exact values for a linear system involving the equation \(x_1 + x_2 + c = x_0\) and two colors: Let \(c\) and \(k\) be integers with \(-1 \leq c \leq k\). Then the 2-color Rado number for the system \[\begin{array}{lcl} x_1 + x_2 + c &=& x_0, \\ y_1 + y_2 + k &=& y_0, \end{array}\] is infinite if \(c\) and \(k\) have opposite parity, and has a value of \(4k+5\) if \(c\) and \(k\) have the same parity. We also extend this to the continuous result where we color the real numbers.

In this article, we begin to investigate prime labelings of the zero-divisor graph of a commutative ring. A graph \(G\) with \(n\) vertices admits a prime labeling if the vertices can be labeled using distinct positive integers less than or equal to \(n\) such that any two adjacent vertices have labels which are relatively prime. We are able to construct several infinite families of commutative rings which will have prime labelings for their zero-divisor graphs. We also find infinite families of commutative rings which do not have prime labelings for their zero-divisor graphs. We then continue the process of determining which commutative rings will have prime labelings for their zero-divisor graphs by resolving the question for all rings with 14 or fewer vertices in their zero-divisor graph. We conclude with several unresolved questions that could be interesting to pursue further.

Some methods of constructions of square tactical decomposable regular group divisible designs are described. These designs are useful in threshold schemes. An L_2 design is also identified as square tactical decomposable. This completes spectrum of the solutions of entire L_2 designs listed in Clatworthy [2] using matrix approaches.

Let \(G\) be a loopless connected graph. A graph \(G\) is reduced if it contains no collapsible subgraph. Catlin (posted by Chen and Lai [9]) conjectured that every connected reduced graph is either 2-colorable or 3-colorable. A weaker conjecture states that the independence number of a connected reduced graph \(G\) is at least one-third of its number of vertices. In this paper, we establish a lower bound on the independence number in reduced graphs. As an application, we examine the independence number conjecture for reduced graphs with a given upper bound on the number of vertices. Also, we investigate the chromatic number of reduced planar graphs under given conditions.

Graph pebbling is a network optimization method modeling the movement of resources in transit. A pebbling move on a connected graph \(G\) removes two pebbles from a vertex, places one on an adjacent vertex, and discards the other, with the loss analogous to packet loss in communication networks. The generalized version, \(t\)-pebbling, defines the \(t\)-pebbling number \(f_t(G)\) as the smallest integer such that, from any distribution of \(f_t(G)\) pebbles, \(t\) pebbles can be moved to any vertex \(v\) via a pebbling sequence. A graph satisfies the \(2t\)-pebbling property if \(2t\) pebbles can be transferred to \(v\) when \(2f_t(G)-q+1\) pebbles are distributed, where \(q\) is the number of occupied vertices. This paper establishes a lower bound for the rooted product of two graphs \(G\) and \(H\), sharp when one factor is a path, complete graph, or star. Further results on pebbling in triangle-free graphs are also obtained, including verification of the \(2t\)-pebbling property for rooted products involving such graphs.

It is known that null graphs are the only (regular) graphs with local antimagic chromatic 1 and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we first use matrices of size \((2m+1) \times (2k+1)\) to completely determine the local antimagic chromatic number of the join of null graphs \(O_m\) and 1-regular graphs \((2k+1)P_2\) for all \(k\ge 1, m\ge 2\). We then make use of other matrices of same size to obtain the local antimagic chromatic number of another family of tripartite graphs. Consequently, we obtained infinitely many (possibly disconnected) regular tripartite graphs with local antimagic chromatic number 3.

Let \(2\le k\in\mathbb{Z}\). We say that a total coloring of a \(k\)-regular simple graph via \(k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. We prove that Hamming shells of star transposition graphs and Hamming cubes have efficient total colorings. Also in this work, a focus is set upon the graphs of girths \(2k\) and \(k\). Efficient total colorings of finite connected simple cubic graphs of girth 6 are constructed. These are of two specific types, namely: (a) those whose 6-cycles use just 3 colors with antipodal monochromatic pairs of vertices or edges; (b) those whose 6-cycles do not respect item (a) so they use four colors. An orthogonality property holds for all graphs of type (a). Such orthogonality property allows further edge-half-girth colorings in the corresponding prism graphs.

A radio labeling of a graph \( G \) is a mapping \( f : V(G) \to \{0, 1, 2, \dots\} \) such that \( |f(u)-f(v)| \geq \text{diam}(G) + 1 – d(u,v) \) for every pair of distinct vertices \( u,v \) of \( G \), where \( \text{diam}(G) \) is the diameter of \( G \) and \( d(u,v) \) is the distance between \( u \) and \( v \) in \( G \). The radio number \( \text{rn}(G) \) of \( G \) is the smallest integer \( k \) such that \( G \) admits a radio labeling \( f \) with \( \max\{f(v) : v \in V(G)\} = k \). In this paper, we give a lower bound for the radio number of the Cartesian product of a tree and a complete graph and give two necessary and sufficient conditions for the sharpness of the lower bound. We also give three sufficient conditions for the sharpness of the lower bound. We determine the radio number of the Cartesian product of a level-wise regular tree and a complete graph which attains the lower bound. The radio number of the Cartesian product of a path and a complete graph derived in [B. M. Kim, W. Hwang, and B. C. Song, Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139–149] can be obtained using our results in a short way.

Let \( G \) be a connected graph with \( m \) edges. The density of a nontrivial subgraph \( H \) with \( \omega(H) \) components is \( d(H) = \frac{|E(H)|}{|V(H)| – \omega(H)} \). A graph \( G \) is uniformly dense if for any nontrivial subgraph \( H \) of \( G \), \( d(H) \leq d(G) \). For each cyclic ordering \( o=(e_1, e_2, \dots, e_m) \) of \( E(G) \), let \( h(o) \) be the largest integer \( k \) such that every \( k \) cyclically consecutive elements in \( o \) induce a forest in \( G \); and the largest \( h(o) \), taken among all cyclic orderings of \( G \), is denoted by \( h(G) \). A cyclic ordering \( o \) of \( G \) is a cyclic base ordering if \( h(o) = |V(G)| – \omega(G) \). In [15], Kajitani et al. proved that every connected nontrivial graph with a cyclic base ordering is uniformly dense, and conjectured that every uniformly dense graph has a cyclic base ordering. This motivates the study of \( h(G) \). In this paper, we investigate the value of \( h \) for some families of graphs and determine all connected graphs \( G \) with \( h(G) \leq 2 \).