Discrete Mathematics最新文献

We consider finite models of the first group of Hilbert's axioms of the Euclidean geometry (the so-called axioms of incidence). We give a lower bound on the number of such models with n points, and we calculate their exact number for n up to 12.

For a digraph Γ, if F is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of Γ, then F is called the splitting field of Γ. The extension degree of F over the field of rational numbers $Q$ is said to be the algebraic degree of Γ. A digraph is a semi-Cayley digraph over a group G if it admits G as a semiregular automorphism group with two orbits of equal size. A semi-Cayley digraph $\mathrm{SC}(G,T_{11},T_{22},T_{12},T_{21})$ is called quasi-abelian if each of $T_{11},T_{22},T_{12}$ and $T_{21}$ is a union of some conjugacy classes of G. This paper determines the splitting field and the algebraic degree of a quasi-abelian semi-Cayley digraph over any finite group in terms of irreducible characters of groups. This work generalizes the previous works on algebraic degrees of Cayley graphs over abelian groups and any group having a subgroup of index 2, and semi-Cayley digraphs over abelian groups.

A proper n-coloring of a graph G is an assignment of colors from $\{1,\dots ,n\}$ to its vertices such that no two adjacent vertices get assigned the same color. The chromatic number of G, denoted by $\chi \left(G\right)$, refers to the smallest n such that G admits a proper n-coloring. This notion naturally extends to edge-colorings (resp. total-colorings) when edges (resp. both vertices and edges) are to be colored, and this provides other parameters of G: its chromatic index ${\chi }^{\prime }\left(G\right)$ and its total chromatic number ${\chi }^{″}\left(G\right)$.

These coloring notions are among the most fundamental ones of the graph coloring theory. As such, they gave birth to hundreds of studies dedicated to several of their aspects, including generalizations to more general structures such as oriented graphs. They include notably the notions of oriented n-colorings and oriented n-arc-colorings, which stand as natural extensions of their undirected counterparts, and which have been receiving increasing attention.

Our goal is to introduce a missing piece in this line of work, namely the oriented counterparts of proper n-total-colorings and total chromatic number. We first define these notions and show that they share properties and connections with oriented (arc) colorings that are reminiscent of those shared by their undirected counterparts. We then focus on understanding the oriented total chromatic number of particular types of oriented graphs, such as oriented forests, cycles, and some planar graphs. Finally, we establish a full complexity dichotomy for the problem of determining whether an oriented graph is totally k-colorable.

Throughout this work, each of our results is compared to what is known regarding the oriented chromatic number and oriented chromatic index. We also disseminate some directions for further research on the oriented total chromatic number.

In this work we introduce new combinatorial objects called d–fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d\left(n\right)$ to be their counting function. We also consider the Schmidt type d–fold partition diamonds, which have counting function $s_d\left(n\right)$. Using partition analysis, we then find the generating function for both, and connect the generating functions ${\sum }_{n=0}^{\infty }{s}_d\left(n\right)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan–like congruences satisfied by $s_d\left(n\right)$ for various values of d, including the following family: for all $d\ge 1$ and all $n\ge 0$, $s_d(2n+1)\equiv 0\left(\mathrm{mod}{2}^d\right)$.

Stanley's Tree Isomorphism Conjecture posits that the chromatic symmetric function can distinguish non-isomorphic trees. This conjecture is already established for caterpillars and other subclasses of trees. We prove the conjecture's validity for a new class of trees that generalize proper caterpillars, thus confirming the conjecture for a broader class of trees.

A known Alder-type partition inequality of level a, which involves the second Rogers–Ramanujan identity when the level a is 2, states that the number of partitions of n into parts differing by at least d with the smallest part being at least a is greater than or equal to that of partitions of n into parts congruent to $\pm a(\mathrm{mod}d+3)$, excluding the part $d+3-a$. In this paper, we prove that for all values of d with a finite number of exceptions, an arbitrary level a Alder-type partition inequality holds without requiring the exclusion of the part $d+3-a$ in the latter partition.

The gem is the 5-vertex graph consisting of a 4-vertex path plus a vertex adjacent to each vertex of the path. A graph is said to be gem-free if it does not contain gem as a subgraph. In this paper, we consider the spectral extremal problem for gem-free graphs with given size. The maximum spectral radius of gem-free graphs with size $m\ge 11$ is obtained, and the unique corresponding extremal graph is determined.

We introduce a closure technique for Hamilton-connectedness of $\{K_{1,3},{\Gamma }_3\}$-free graphs, where ${\Gamma }_3$ is the graph obtained by joining two vertex-disjoint triangles with a path of length 3. The closure turns a claw-free graph into a line graph of a multigraph while preserving its (non)-Hamilton-connectedness. The most technical parts of the proof are computer-assisted.

The main application of the closure is given in a subsequent paper showing that every 3-connected $\{K_{1,3},{\Gamma }_3\}$-free graph is Hamilton-connected, thus resolving one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness.

Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.