Dong, F. M.

Let G be a connected graph with vertex set V (G). A set S of vertices in G is called a weakly connected dominating set of G if (i) S is a dominating set of G and (ii) the graph obtained from G by removing all edges joining two vertices in V (G) \ S is connected. A weakly connected dominating set S of G is said to be minimum or a γw-set if |S| is minimum among all weakly connected dominating sets of G. We say that G is γw-unique if it has a unique γw-set. Recently, a constructive characterisation of γw-unique trees was obtained by Lemanska and Raczek [Czechoslovak Math. J. 59 (134) (2009), 95–100]. A graph is said to be cycle-disjoint if no two cycles in G have a vertex in common. In this paper, we extend the above result on trees by establishing a constructive characterisation of γw-unique cycle-disjoint graphs.

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial Σnk =1α(G, k)(x)k of degree n, where (x)k = x(x−1) . . . (x−k+1) and α(G, k) is real for all k. The adjoint polynomial of G is defined to be Σnk=1α(G, k)μk, where G is the complement of G. We find the zeros of the adjoint polynomials of paths and cycles.

Let G = (V, E) be a simple graph with n vertices and m edges, P (G, k) be the chromatic polynomial of G, and P (G, L) be the number of L-colorings of G for any k -assignment L. In this article, we show that when k ≥ m ‒ 1 ≥ 3, P (G, L) ‒ P (G, k) is bounded below by ((k - m + 1)k ^{n - 3} + ^{(k - m + 3) c}/₃ k ^{n - 5}) ^∑_{u v ∈ E} ∣ L(u) ∖ L(v)∣, where c ≥ ^{(m - 1)(m - 3)} / ₈, and in particular, if G is K₃ -free, then c ≥ (^{m - 2} ₂) + 2 √ m -3. Consequently, P (G, L) ≥P (G, k) whenever k ≥ m - 1.

A 1-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing. Czap and Hudák showed that every 1-planar graph with n vertices has crossing number at most n−2. In this paper, we prove that every maximal 1-planar graph G with n vertices has crossing number at most n−2−(2λ1+2λ2+λ3)/6, where λ1 and λ2 are respectively the numbers of 2-degree and 4-degree vertices in G, and λ3 is the number of odd vertices w in G such that either dG(w)≤9 or G−w is 2-connected. Furthermore, we show that every 3-connected maximal 1-planar graph with n vertices and m edges has crossing number m−3n+6.

For a connected graph G and any non-empty S ⊆ V (G), S is called a weakly connected dominating set of G if the subgraph obtained from G by removing all edges each joining any two vertices in V (G) \ S is connected. The weakly connected domination number γw(G) is defined to be the minimum integer k with |S| = k for some weakly connected dominating set S of G. In this note, we extend a result on the lower bound for the weakly connected domination number γw(G) on trees to cycle-e-disjoint graphs, i.e., graphs in which no cycles share a common edge. More specifically, we show that if G is a connected cycle-e-disjoint graph, then γw(G) ≥ (|V (G)| − v1(G) − nc(G) − noc(G) + 1)/2, where nc(G) is the number of cycles in G, noc(G) is the number of odd cycles in G and v1(G) is the number of vertices of degree 1 in G. The graphs for which equality holds are also characterised.

Let G be a complete multi-partite graph of order n. In this paper, we consider the anti-Ramsey number ar(G, Tq) with respect to G and the set Tq of trees with q edges, where 2 ≤ q ≤ n − 1. For the case q = n − 1, the result has been obtained by Lu, Meier and Wang. We will extend it to q < n−1. We first show that ar(G, Tq) = ℓq(G)+1, where ℓq(G) is the maximum size of a disconnected spanning subgraph H of G with the property that any two components of H together have at most q vertices. Using this equality, we obtain the exact values of ar(G, Tq) for n − 3 ≤ q ≤ n − 1. Moreover, for the general case when (4n − 2)/5 ≤ q ≤ n − 1, ar(G, Tq) can be determined by a simple algorithm. In particular, the explicit expression of ar(G, Tq) is given when G has a partite set much larger than all the other partite sets.

This note presents two results on real zeros of chromatic polynomials. The first result states that if G is a graph containing a q-tree as a spanning subgraph, then the chromatic polynomial P(G, λ) of G has no non-integer zeros in the interval (0, q). Sokal conjectured that for any graph G and any real λ > Δ(G), P(G, λ) > 0. Our second result confirms that it is true if Δ(G) ≥ [n/3] − 1, where n is the order of G.

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

A signed edge domination function (or SEDF) of a simple graph G=(V,E) is a function f:E→{1,−1} such that ∑e′∈N[e]f(e′)≥1 holds for each edge e∈E, where N[e] is the set of edges in that share at least one endpoint with e. Let γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved for the two cases vodd(G)=0 and veven(G)=0, where vodd(G) (resp. veven(G)) is the number of odd (resp. even) vertices in G. This article proves Xu’s conjecture for veven(G)∈{1,2}. We also show that for any simple graph G of order n, γs′(G)≤n+vodd(G)∕2 and γs′(G)≤n−2+veven(G) when veven(G)>0, and thus γs′(G)≤(4n−2)∕3. Our result improves the best current upper bound of γs′(G)≤⌈3n∕2⌉.