On considère un graphe complet dont les arêtes sont totalement préordonnées. En analyse de similitude, plutôt que de procéder à un ordonnancement des arêtes ex oequo par une méthode lexicographique sur leurs intitulés, l'auteur propose de rechercher la réunion des arbres maximaux (RAM).

Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_{rn+i}/q_{rn+i}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...

Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form $p_{rn+i}/q_{rn+i}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

To study the block structure of a connected graph G = (V,E), we introduce two algebraic approaches that reflect this structure: a binary operation + called a leap operation and a ternary relation L called a leap system, both on a finite, nonempty set V. These algebraic structures are easily studied by considering their underlying graphs, which turn out to be block graphs. Conversely, we define the operation ${+}_G$ as well as the set of leaps $L_G$ of the connected graph G. The underlying graph of ${+}_G$, as well...

Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p/6]}\left(t\right)\equiv -(3/p){\sum }_{x=0}^{p-1}((x³-3x+2t)/p)\left(modp\right)$ and $\left({\sum }_{x=0}^{p-1}((x³+mx+n)/p)\right)²\equiv ((-3m)/p){\sum }_{k=0}^{[p/6]}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right){((4m³+27n²)/\left(12³·4m³\right))}^k\left(modp\right)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right)/m^k\left(modp²\right)$, where m is an integer not divisible by p.

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_i$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m() vertices...

A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains just the...

A graph H is defined to be light in a graph family 𝒢 if there exist finite numbers φ(H, 𝒢) and w(H, 𝒢) such that each G ∈ 𝒢 which contains H as a subgraph, also contains its isomorphic copy K with ΔG(K) ≤ φ(H, 𝒢) and ∑x∈V(K) degG(x) ≤ w(H, 𝒢). In this paper, we investigate light graphs in families of plane graphs of minimum degree 2 with prescribed girth and no adjacent 2-vertices, specifying several necessary conditions for their lightness and providing sharp bounds on φ and w for light K1,3...