A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Chromatic numbers and products

Let f(n) be the smallest number so that there are two n chromatic graphs whose product has chromatic number f(n). Under the assumption that a certain sharper result than one obtained by Duffus et al. [On the chromatic number of the product of graphs, J. Graph Theory 9 (1985) 487–495], and Welzl [Symmetric graphs and interpretations, J. Combin. Theory, Sci. B 37 (1984) 235–244], holds we will prove that f(n)n/2.     

On anti-magic labeling for graph products

An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1,2,…,q} such that the vertex sums are pairwise distinct, where the vertex sum at one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K₂ are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(logp). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.     

On the products arising from the Kummer conjecture

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q.In this paper,we extend the product ∏χ(-1)=-1L(1,χ) with prime q,arising from the Kummer conjecture,to the products of some general Dirichlet series,and give some meaningful estimates for them.     

Adaptable chromatic number of graph products

A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1,2,…,k admits an adapted vertex-colouring of G by the same colours 1,2,…,k. (The adaptable chromatic number is just one more than a previously investigated notion of chromatic capacity.) The adaptable chromatic number of a graph G is smaller than or equal to the ordinary chromatic number of G. While the ordinary chromatic number of all (categorical) powers G^k of G remains the same as that of G, the adaptable chromatic number of G^k may increase with k. We conjecture that for all sufficiently large k the adaptable chromatic number of G^k equals the chromatic number of G. When G is complete, we prove this conjecture with k≥4, and offer additional evidence suggesting it may hold with k≥2. We also discuss other products and propose several open problems.     

Vizing’s conjecture for chordal graphs

Vizing conjectured that γ(G□H)≥γ(G)γ(H) for every pair G,H of graphs, where “□” is the Cartesian product, and γ(G) is the domination number of the graph G. Denote by γⁱ(G) the maximum, over all independent sets I in G, of the minimal number of vertices needed to dominate I. We prove that γ(G□H)≥γⁱ(G)γ(H). Since for chordal graphs γⁱ=γ, this proves Vizing’s conjecture when G is chordal.     

Poncelet's theorem, Sendov's conjecture, and Blaschke products

By connecting Blaschke products, unitary dilations of matrices, numerical range, Poncelet's theorem and interpolation, we extend and simplify Gau and Wu's work (Gau and Wu (2004) [10]). We look at Blaschke products' role in the Sendov conjecture.     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K_p and K_q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4.     

On the relations of graph parameters and its total parameters

In this paper we get some relations between α（G）, α＇（G）, β（G）, β＇（G） and αT（G）, βT（G）. And all bounds in these relations are best possible, where α（G）, α＇（G）,/3（G）, β（G）, αT（G） and βT（G） are the covering number, edge-covering number, independent number, edge-independent number （or matching number）, total covering number and total independent number, respectively.     

On pitfalls in computing the geodetic number of a graph

Given a simple connected graph G = (V, E) the geodetic closure of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes . The geodetic number, denoted by g(G), is the smallest cardinality of a node set S ^* such that I[S ^*] = V. In “The geodetic number of a graph”, [Harary et al. in Math. Comput. Model. 17:89–95, 1993] propose an incorrect algorithm to find the geodetic number of a graph G. We provide counterexamples and show why the proposed approach must fail. We then develop a 0–1 integer programming model to find the geodetic number. Computational results are given.     

On the critical graph conjecture

Gol'dberg has recently constructed an infinite family of 3-critical graphs of even order. We now prove that if there exists a p(≥4)-critical graph K of odd order such that K has a vertex u of valency 2 and another vertex v ≠ u of valency ≤(p + 2)/2, then there exists a p-critical graph of even order.     

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n=|F|?2 and for a∈A let s(a) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a∈A with n−2s(a)?0. Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280-288] we verify the conjecture for the particular case when m?3 and n?m²−²m/2. Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n−2s(a), for all a∈A, is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.