24 self-inscribed decagons in Desargues configuration
2-adic behavior of numbers of domino tilings.
2-distance 4-colorability of planar subcubic graphs with girth at least 22
The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.
2-distance coloring of sparse planar graphs.
2-enumerations of halved alternating sign matrices.
2-factors in claw-free graphs
We consider the question of the range of the number of cycles possible in a 2-factor of a 2-connected claw-free graph with sufficiently high minimum degree. (By claw-free we mean the graph has no induced .) In particular, we show that for such a graph G of order n ≥ 51 with δ(G) ≥ (n-2)/3, G contains a 2-factor with exactly k cycles, for 1 ≤ k ≤ (n-24)/3. We also show that this result is sharp in the sense that if we lower δ(G), we cannot obtain the full range of values for k.
2-factors in claw-free graphs with locally disconnected vertices
An edge of is singular if it does not lie on any triangle of ; otherwise, it is non-singular. A vertex of a graph is called locally connected if the induced subgraph by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex of degree at least in there is a nonnegative integer such that lies...
2-halvable complete 4-partite graphs
A complete 4-partite graph is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs with at most one odd part all d-halvable graphs are known. In the class of biregular graphs with four odd parts (i.e., the graphs and ) all d-halvable graphs are known as well, except for the graphs when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs with three or four different...
2-layer straightline crossing minimization: Performance of exact and heuristic algorithms.
2-placement of (p,q)-trees
Let G = (L,R;E) be a bipartite graph such that V(G) = L∪R, |L| = p and |R| = q. G is called (p,q)-tree if G is connected and |E(G)| = p+q-1. Let G = (L,R;E) and H = (L',R';E') be two (p,q)-tree. A bijection f:L ∪ R → L' ∪ R' is said to be a biplacement of G and H if f(L) = L' and f(x)f(y) ∉ E' for every edge xy of G. A biplacement of G and its copy is called 2-placement of G. A bipartite graph G is 2-placeable if G has a 2-placement. In this paper we give all (p,q)-trees which...
2-recognizability by prime graph of .
2-Tone Colorings in Graph Products
A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number...
3 arten von Linearverbindungen bei Bernoullipolynomen
-configurations with simple edge basis and their corresponding quasigroup identities
There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.
321-polygon-avoiding permutations and Chebyshev polynomials.
3-consecutive c-colorings of graphs
A 3-consecutive C-coloring of a graph G = (V,E) is a mapping φ:V → ℕ such that every path on three vertices has at most two colors. We prove general estimates on the maximum number of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with for k = 3 and k = 4.
3-designs from PGL.
'3in1' enhanced: three squared ways to '3in1' GRAPHS
3-Paths in Graphs with Bounded Average Degree
In this paper we study the existence of unavoidable paths on three vertices in sparse graphs. A path uvw on three vertices u, v, and w is of type (i, j, k) if the degree of u (respectively v, w) is at most i (respectively j, k). We prove that every graph with minimum degree at least 2 and average degree strictly less than m contains a path of one of the types [...] Moreover, no parameter of this description can be improved.
3-polytopes of constant tolerance of edges