We provide a new proof of a result of Hanson and Toft classifying the maximum-size \(K_r\)-free graphs on \(n\) vertices with chromatic number at least \(r\).

Much research has been done on the edge decomposition of \(\lambda\) copies of the complete graph \(G\) with respect to some specified subgraph \(H\) of \(G\). This is equivalent to the investigation of \((G, H)\)-designs of index \(\lambda\). In this paper, we present a fundamental theorem on the decomposition of \(\lambda\) copies of a complete bipartite graph. As an application of this result, we show that necessary conditions are sufficient for the decomposition of \(\lambda\) copies of a complete bipartite graph into several multi-subgraphs \(H\) with a number of vertices less than or equal to \(4\) and the number of edges less than or equal to \(4\), with some exceptions where decompositions do not exist. These decomposition problems are interesting to study as various decompositions do not exist even when necessary conditions are satisfied.

If the integer \(r \geq 2\), say that a composition of the natural number \(n\) is \(r\)-\({regular}\) if no part is divisible by \(r\). Let \(c_r(n)\) denote the number of \(r\)-regular compositions of \(n\) (with \(c_r(0) = 1\)). We show that \(c_r(n)\) satisfies a linear recurrence of order \(r\). We also obtain asymptotic estimates for \(c_r(n)\), and we evaluate \(c_r(n)\) for \(2 \leq r \leq 5\) and \(1 \leq n \leq 10\).

Using only the skein relation and some combinatorics, we find a closed form for the Conway polynomial of \((m,3)\) torus links and a trio of recurrence relations that define the Conway polynomial of any \((m,4)\) torus link.

For positive integers \(c\) and \(d\), let \(K_{c\times d}\) denote the complete multipartite graph with \(c\) parts, each containing \(d\) vertices. Let \(G\) with \(n\) edges be the union of two vertex-disjoint even cycles. We use graph labelings to show that there exists a cyclic \(G\)-decomposition of \(K_{(2n+1)\times t}\), \(K_{(n/2+1)\times 4t}\), \(K_{5\times (n/2)t}\), and of \(K_{2\times 2nt}\) for every positive integer \(t\). If \(n \equiv 0 \pmod{4}\), then there also exists a cyclic \(G\)-decomposition of \(K_{(n+1)\times 2t}\), \(K_{(n/4+1)\times 8t}\), \(K_{9\times (n/4)t}\), and of \(K_{3\times nt}\) for every positive integer \(t\).

For a Hamiltonian graph \(G\), the Hamiltonian cycle extension number of \(G\) is the maximum positive integer \(k\) for which every path of order \(k\) or less is a subpath of some Hamiltonian cycle of \(G\). The Hamiltonian cycle extension numbers of all Hamiltonian complete multipartite graphs are determined. Sharp lower bounds for the Hamiltonian cycle extension number of a Hamiltonian graph are presented in terms of its minimum degree and order, its size and the sum of the degrees of every two non-adjacent vertices. Hamiltonian cycle extension numbers are also determined for powers of cycles.

We give conditions on the numbers \(\{\varphi_{ij}\}\) under which a vertex-degree-based topological index \(TI\) of the form

\[
TI(G) = \sum_{1\leq i\leq j\leq n-1} m_{ij}\varphi_{ij},
\]

where \(G\) is a graph with \(n\) vertices and \(m_{ij}\) is the number of \(ij\)-edges, has the zigzag chain as an extreme value among all polyomino chains. As a consequence, we deduce that over the polyomino chains, the zigzag chain has the maximal value of the Randić index, the sum-connectivity index, the harmonic index, and the geometric-arithmetic index, and the minimal value of the first Zagreb index, second Zagreb index, and atom-bond-connectivity index.

A cluster of \( 2n+1 \) cubes comprising the central cube and reflections in all its faces is called the \( n \)-dimensional cube. If \( 2n+1 \) is not a prime, then there are infinitely many tilings of \( \mathbb{R}^n \) by crosses, but it has been conjectured that there is a unique tiling of \( \mathbb{R}^n \) by crosses otherwise. The conjecture has been proved for \( n=2,3 \), and in this paper, we prove it also for \( n=5 \). So there is a unique tiling of \( \mathbb{R}^3 \) by crosses, there are infinitely many tilings of \( \mathbb{R}^4 \), but for \( \mathbb{R}^5 \), there is again only one tiling by crosses. We consider this result to be a paradox as our intuition suggests that “the higher the dimension of the space, the more freedom we get.
“`

A graph \( G \) is collapsible if for every even subset \( R \subseteq V(G) \), there is a spanning connected subgraph \( H_R \) of \( G \) whose set of odd degree vertices is \( R \). A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph \( G \) can be determined by the reduced graph obtained from \( G \) by contracting all the collapsible subgraphs of \( G \). In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph \( G \) of order \( n \) either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if \( G \) satisfies one of the following:

1. \( d(u) + d(v) > 2\left(\frac{n}{15} – 1\right) \) for any \( uv \notin E(G) \) and \( n \) is large;
2. the size of a maximum matching in \( G \) is at most 6;
3. the independence number of \( G \) is at most 5.

These are improvements of prior results in [16], [18], [24], and [25].

In this note, we consider the lexicographical ordering by spectral moments of trees with a given degree sequence. Such questions have been studied for a variety of different categories of trees. Particularly, the last tree in this ordering among trees with a given degree sequence was recently identified in two independent manuscripts. The characterization of the first such trees, however, remains open. We make some progress on this question in this note, by making use of the interpretation of the spectral moment in terms of numbers of paths and the product of adjacent vertex degrees, the first trees are characterized with the additional condition that the nonleaf vertex degrees are different from each other. We also comment on the case when there are repetitions in the vertex degrees.