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.

We determine all 120 nonisomorphic systems obtainable from the projective Steiner triple system of order 31 by at most three Pasch trades. Exactly three of these, each corresponding to three Pasch trades, are rigid. Thus three Pasch trades suffice, and are required, in
order to convert the projective system of order 31 to a rigid system. This contrasts with the projective system of order 15 where four Pasch trades are required. We also show that four Pasch trades are required in order to convert the projective system of order 63 to a
rigid system.

In the paper “Eternal security in graphs” by Goddard, Hedetniemi and Hedetniemi (2005, [4]), the authors claimed that, for any Cayley graph, the eternal \(m\)-security number equals the minimum cardinality of a dominating set. However, the equality is false. In this note, we present a counterexample and comment on the eternal \(m\)-security number for Cayley graphs.

Define \(\mathcal{D}_k\) to be the class of graphs such that, for every independent set \(\{v_1,\ldots,v_h\}\) of vertices with \(2 \leq h \leq k\), if \(S\) is an inclusion-minimal set of vertices whose deletion would put \(v_1,\ldots,v_h\) into \(h\) distinct connected components, then \(S\) induces a complete subgraph; also, let \(\mathcal{D} = \bigcap_{k\geq2} \mathcal{D}_k\). Similarly, define \(\mathcal{D}_k’\) and \(\mathcal{D}’\) with “complete” replaced by “edgeless,” and define \(\mathcal{D}_k^*\) and \(\mathcal{D}^*\) with “complete” replaced by “complete or edgeless.” The class \(\mathcal{D}_2\) is the class of chordal graphs, and the classes \(\mathcal{D}\), \(\mathcal{D}_2’\), and \(\mathcal{D}_2^*\) have also been characterized recently. The present paper gives unified characterizations of all of the classes \(\mathcal{D}_k\), \(\mathcal{D}_k’\), \(\mathcal{D}_k^*\), \(\mathcal{D}\), \(\mathcal{D}’\), and \(\mathcal{D}^*\).

Catalan Numbers and their generalizations are found throughout the field of Combinatorics. This paper describes their connection to numbers whose digits appear in the number’s own \(p^{th}\) root. These are called Grafting Numbers and they are defined by a class of polynomials given by the Grafting Equation: \((x+y)^p = b^ax\). A formula that solves for \(x\) in these polynomials uses a novel extension to Catalan Numbers and will be proved in this paper. This extension results in new sequences that also solve natural extensions to previous Combinatorics problems. In addition, this paper will present computationally verified conjectures about formulas and properties of other solutions to the Grafting Equation.

A \( d \)-angulation of a surface is an embedding of a 3-connected graph on that surface that divides it into \( d \)-gonal faces. A \( d \)-angulation is said to be Grünbaum colorable if its edges can be \( d \)-colored so that every face uses all \( d \) colors. Up to now, the concept of Grünbaum coloring has been related only to triangulations (\( d = 3 \)), but in this note, this concept is generalized for an arbitrary face size \( d \geq 3 \). It is shown that the face 2-colorability of a \( d \)-angulation \( P \) implies the Grünbaum colorability of \( P \). Some wide classes of triangulations have turned out to be face 2-colorable.