Let \(C_n^{(t)}\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(n = 3, 5, 6, 7, 9, 4k\). In this paper, the conjecture is shown to be true for \(n = 11\).

Let \(P(G; \lambda)\) denote the chromatic polynomial of a graph \(G\), expressed in the variable \(\lambda\). Then \(G\) is said to be chromatically unique if \(G\) is isomorphic with \(H\) for any graph \(H\) such that \(P(H; \lambda) = P(G; \lambda)\). The graph consisting of \(s\) edge-disjoint paths joining two vertices is called an \(s\)-bridge graph. In this paper, we provide a new family of chromatically unique \(5\)-bridge graphs.

Recently in \([5]\), the author considered certain reciprocal sums of general second order recurrence \(\{W_n\}\). In this paper, we generalize the results of Xi and we give some new results for the reciprocal sums of \(k\)-th power of general second order recurrence \(\{W_{kn}\}\) for arbitrary positive integer \(k\).

In this article, we study the generalized Bernoulli and Euler polynomials, and obtain relationships between them, based upon the technique of matrix representation.

Let \(K_v\) be the complete graph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-decomposition of \(K_v\), denoted by \(G\)-GD\((v)\), is a pair \((X, \mathcal{B})\) where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, nine graphs \(G_i\) with six vertices and nine edges are discussed, and the existence of \(G_i\)-decompositions are completely solved, \(1 \leq i \leq 9\).

A graph \(G\) is called \({uniquely\; k-list \;colorable}\), or \(UkLC\) for short, if it admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring. A graph \(G\) is said to have the property \(M(k)\) (M for Marshal Hall) if and only if it is not \(UkLC\). In \(1999\), M. Ghebleh and E.S. Mahmoodian characterized the \(U3LC\) graphs for complete multipartite graphs except for nine graphs. At the same time, for the nine exempted graphs, they give an open problem: verify the property \(M(3)\) for the graphs \(K_{2,2,\ldots,2}\) for \(r = 4,5,\ldots,8\), \(K_{2,3,4}\), \(K_{1*4,4}\), \(K_{1*4,4}\), and \(K_{1*5,4}\). Until now, except for \(K_{1*5,4}\), the other eight graphs have been showed to have the property \(M(3)\) by W. He et al. In this paper, we show that graph \(K_{1*5,4}\) has the property \(M(3)\), and as consequences, \(K_{1*4,4}\), \(K_{2,2,4}\) have the property \(M(3)\). Therefore the \(U3LC\) complete multipartite graphs are completely characterized.

Given a graph \(G\), we say \(S \subseteq V(G)\) is \({resolving}\) if for each pair of distinct \(u, v \in V(G)\) there is a vertex \(x \in S\) where \(d(u, x) \neq d(v, x)\). The metric dimension of \(G\) is the minimum cardinality of all resolving sets. For \(w \in V(G)\), the distance from \(w\) to \(S\), denoted \(d(w, S)\), is the minimum distance between \(w\) and the vertices of \(S\). Given \(\mathcal{P} = \{P_1, P_2, \ldots, P_k\}\) an ordered partition of \(V(G)\), we say \(P\) is resolving if for each pair of distinct \(u, v \in V(G)\) there is a part \(P_i\) where \(d(u, P_i) \neq d(v, P_i)\). The partition dimension is the minimum order of all resolving partitions. In this paper, we consider relationships between metric dimension, partition dimension, diameter, and other graph parameters. We construct “universal examples” of graphs with given partition dimension, and we use these to provide bounds on various graph parameters based on metric and partition dimensions. We form a construction showing that for all integers \(a\) and \(b\) with \(3 \leq a \leq \beta + 1\), there exists a graph \(G\) with partition dimension \(\alpha\) and metric dimension \(\beta\), answering a question of Chartrand, Salehi, and Zhang \([3]\).

The total chromatic number \(\chi_\tau(G)\) is the least number of colours needed to colour the vertices and edges of a graph \(G\) such that no incident or adjacent elements (vertices or edges) receive the same colour. This work determines the total chromatic number of grids, particular cases of partial grids, near-ladders, and of \(k\)-dimensional cubes.

The \({star arboricity}\) \(sa(G)\) of a graph \(G\) is the minimum number of star forests which are needed to decompose all edges of \(G\). For integers \(k\) and \(n\), \(1 \leq k \leq n\), the \({crown}\) \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j : i = 0, 1, \ldots, n-1, j \equiv i+1, i+2, \ldots, i+k \pmod{n}\}\). In \([2]\), Lin et al. conjectured that for every \(k\) and \(n\), \(3 \leq k \leq n-1\), the star arboricity of the crown \(C_{n,k}\) is \(\lceil k/2 \rceil + 1\) if \(k\) is odd and \(\lceil k/2 \rceil + 2\) otherwise. In this note, we show that the above conjecture is not true for the case \(n = 9t\) (\(t\) is a positive integer) and \(k = 4\) by showing that \(sa(C_{9t,4}) = 3\).

Let \(\mathcal{P}(n,k)\) denote the number of graphs on \(n+k\) vertices that contain \(P_n\), a path on \(n\) vertices, as an induced subgraph. In this note, we will find upper and lower bounds for \(\mathcal{P}(n,k)\). Using these bounds, we show that for \(k\) fixed, \(\mathcal{P}(n,k)\) behaves roughly like an exponential function of \(n\) as \(n\) gets large.