Circular chromatic numbers of some distance graphs

AbstractGiven a set D of positive integers, the distance graph G(Z,D) has vertices all integers Z, and two vertices j and j′ in Z are adjacent if and only if |j-j′|∈D. This paper determines the circular chromatic numbers of some distance graphs

On Structure of Some Plane Graphs with Application to Choosability

AbstractA graph G=(V, E) is (x, y)-choosable for integers x>y⩾1 if for any given family {A(v)∣v∈V} of sets A(v) of cardinality x, there exists a collection {B(v)∣v∈V} of subsets B(v)⊂A(v) of cardinality y such that B(u)∩B(v)=∅ whenever uv∈E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k∈{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable

Several parameters of generalized Mycielskians

AbstractThe generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m⩾0, one can transform G into a new graph μm(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μm(G)

The L(2,1)-choosability of cycle

For a given graph $G=(V,E)$, let $mathscr L(G)={L(v) : vin V}$ be a prescribed list assignment. $G$ is $mathscr L$-$L(2,1)$-colorable if there exists a vertex labeling $f$ of $G$ such that $f(v)in L(v)$ for all $v in V$; $|f(u)-f(v)|geq 2$ if $d_G(u,v) = 1$; and $|f(u)-f(v)|geq 1$ if $d_G(u,v)=2$. If $G$ is $mathscr L$-$L(2,1)$-colorable for every list assignment $mathscr L$ with $|L(v)|geq k$ for all $vin V$, then $G$ is said to be $k$-$L(2,1)$-choosable. In this paper, we prove all cycles are $5$-$L(2,1)$-choosable

Duality in the Bandwidth Problem

The bandwidth is an important invariant in graph theory. However, the problem to determine the bandwidth of a general graph is NP-complete. To get sharp bounds, we propose to pay attention to various duality properties or minimax relations related to the bandwidth problem. This paper presents a summary in this point of view

Fully angular hexagonal chains extremal with regard to the largest eigenvalue

1298-1303Let G be a molecular graph with characteristic polynomial ϕ(G,x). The leading eigenvalue of G is the largest root of the equation ϕ(G,x) =0. In this paper, the hexagonal chain (unbranched catacondensed benzenoid molecule) with minimum leading eigenvalue among fully angular hexagonal chains having a given number of hexagons is determined

Edge Game Coloring of Graphs

Corresponding to the game chromatic number of graphs, we consider in this paper the game chromatic index Ø 0 g of graphs, which is defined similarly, except that edges, instead of vertices of graphs are colored. Upper bounds for trees and wheels are given

On Some Three-color Ramsey Numbers

In this paper we study three-color Ramsey numbers. Let Ki,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G1, G2 and G3, if r(G1, G2) ≥ s(G3), then r(G1, G2, G3) ≥ (r(G1, G2) − 1)(χ(G3) − 1) + s(G3), where s(G3) is the chromatic surplus of G3; (ii)(k + m − 2)(n − 1) + 1 ≤ r(K1,k, K1,m, Kn) ≤ (k + m − 1)(n − 1) + 1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m ≥ k ≥ 2, there is a constant c such that r(Kk,m, Kk,m, Kn) ≤ c(n / log n) k, and r(C2m, C2m, Kn) ≤ c(n / log n) m/(m−1) for sufficiently large n