(2,1)-Total labelling of outerplanar graphs

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper we prove that if G is an outerplanar graph with maximum degree Δ(G), then if Δ(G)?5, or Δ(G)=3 and G is 2-connected, or Δ(G)=4 and G contains no intersecting triangles.     

不含5-圈和6-圈的平面图的(2,1)-全标号

图G的(2,1)-全标号是对图G的顶点和边的一个标号分配,使得:(1)任意两个相邻顶点标号不同;(2)任意两条相邻边标号不同;(3)任意顶点与其相关联的边标号至少相差2.两个标号的最大差值称为跨度,图G的所有(2,1)-全标号的最小跨度称为(2,1)-全标号数,记为λ_2~T(G).本文证明了如果G是一个?=p+5的平面图,且G不包含5-圈和6-圈,那么λ_2~T(G)=2?-p,p=1,2,3.     

Polynomial algorithm for sharp upper bound of rainbow connection number of maximal outerplanar graphs

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2011) [5] proved that computing rc(G) is NP-hard and deciding if rc(G)=2 is NP-complete. When edges of G are colored with fixed number k of colors, Kratochvil [6] proposed a question: what is the complexity of deciding whether G is rainbow connected? is this an FPT problem? In this paper, we prove that any maximal outerplanar graph is k rainbow connected for suitably large k and can be given a rainbow coloring in polynomial time.     

A tight upper bound on the cover time for random walks on graphs

We prove that the expected time for a random walk to visit all n vertices of a connected graph is at most 4/27n³ + o(n³).     

Sum coloring and interval graphs: a tight upper bound for the minimum number of colors

The SUM COLORING problem consists of assigning a color c(v_i)Z⁺ to each vertex v_iV of a graph G=(V,E) so that adjacent nodes have different colors and the sum of the c(v_i)'s over all vertices v_iV is minimized. In this note we prove that the number of colors required to attain a minimum valued sum on arbitrary interval graphs does not exceed min{n;2χ(G)−1}. Examples from the papers [Discrete Math. 174 (1999) 125; Algorithmica 23 (1999) 109] show that the bound is tight.     

Game chromatic number of outerplanar graphs

This note proves that the game chromatic number of an outerplanar graph is at most 7. This improves the previous known upper bound of the game chromatic number of outerplanar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 67–70, 1999     

An upper bound for the chromatic number of line graphs

It was conjectured by Reed [B. Reed, ω,α, and χ, Journal of Graph Theory 27 (1998) 177–212] that for any graph G, the graph’s chromatic number χ(G) is bounded above by , where Δ(G) and ω(G) are the maximum degree and clique number of G, respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.     

A tight upper bound for the number of intersections between two rectangular paths

The problem of finding the number of intersections between two geometric figures in the plane has been studied extensively in literature. In this paper, the geometric figure comprising a continuous rectilinear path (called rectangular path) is considered, and a tight (least) upper bound onI(P, Q), the number of intersections between two rectangular pathsP andQ, is given.Editors' Note: One of our referees has reported that the main result of this paper has recently been given independently by a Chinese researcher at the University of Science and Technology, Hefei, P. R. of China. His paper is under publication in the Chinese Science Bulletin. However, since this journal may not be easily accessible to our readers, and further the two papers are obviously independent of each other, theBIT Editors have decided to accept the present paper.     

Acyclic edge chromatic number of outerplanar graphs

A proper edge coloring of a graph G is called acyclic if there is no 2‐colored cycle in G. The acyclic edge chromatic number of G, denoted by χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

Proper orientation number of triangle-free bridgeless outerplanar graphs

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation proper if neighboring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\overrightarrow{\chi }\left(G\right)$, is the minimum maximum in-degree of a proper orientation of $G$. Araujo et al asked whether there is a constant $c$ such that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le c$ for every outerplanar graph $G$ and showed that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 7$ for every cactus $G$. We prove that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 3$ if $G$ is a triangle-free 2-connected outerplanar graph and $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 4$ if $G$ is a triangle-free bridgeless outerplanar graph.     

A new upper bound for the independence number of edge chromatic critical graphs

In 1968, Vizing conjectured that if G is a Δ‐critical graph with n vertices, then α(G)≤n/2, where α(G) is the independence number of G. In this paper, we apply Vizing and Vizing‐like adjacency lemmas to this problem and prove that α(G)<(((5Δ?6)n)/(8Δ?6))<5n/8 if Δ≥6. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 202‐212, 2011     

CHROMATIC NUMBER OF SQUARE OF MAXIMAL OUTERPLANAR GRAPHS

Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ ＋ 1 ≤ x（G^2） ≤△ ＋ 2, and x（G^2） = A ＋ 2 if and only if G is Q, where A represents the maximum degree of G.     

The relaxed game chromatic number of outerplanar graphs

The (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004     

An upper bound for the total chromatic number of dense graphs

In this paper we consider those graphs that have maximum degree at least 1/k times their order, where k is a (small) positive integer. A result of Hajnal and Szemerédi concerning equitable vertex-colorings and an adaptation of the standard proof of Vizing's Theorem are used to show that if the maximum degree of a graph G satisfies Δ(G) ≥ |V(G)/k, then X″(G) ≤ Δ(G) + 2k + 1. This upper bound is an improvement on the currently available upper bounds for dense graphs having large order.     

An upper bound for domination number of 5-regular graphs

Let G = (V, E) be a simple graph. A subset S ⊆ V is a dominating set of G, if for any vertex u ∈ V-S, there exists a vertex v ∈ S such that uv ∈ E. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. In this paper we will prove that if G is a 5-regular graph, then γ(G) ⩽ 5/14n.     

On the minimal reducible bound for outerplanar and planar graphs

Let L be the set of all additive and hereditary properties of graphs. For P₁, P₂ L we define the reducible property R = P₁ P₂ as follows: G P₁P₂ if there is a bipartition (V₁, V₂) of V(G) such that V₁ P₁ and V₂ P₂. For a property P L, a reducible property R is called a minimal reducible bound for P if P R and for each reducible property R′, R′ R → P R′. It is proved that the class of all outerplanar graphs has exactly two minimal reducible bounds in L. Some related problems for planar graphs are discussed.     

A note on construction of upper bound graphs

In this paper, we consider construction of upper bound graphs. An upper bound graph can be transformed into a nova by contractions and a nova can be transformed into an upper bound graph by splits. These results include a characterization on upper bound graphs.