Dong, F. M.

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⌉.

For any graph 𝘎, the chromatic polynomial of 𝘎 is the function 𝘗 (𝘎, 𝓂) which counts the number of proper m-colorings of 𝘎 for each positive integer m. The DP color function 𝘗ᴅᴘ(𝘎, 𝓂) of 𝘎, introduced by Kaul and Mudrock in 2019, is a generalization of 𝘗 (𝘎, 𝓂) with 𝘗DP(𝘎, 𝓂)≤ 𝘗 (𝘎, 𝓂) for each positive integer 𝓂. Let 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎)) denote the property that 𝘗ᴅᴘ(𝘎, 𝓂)= 𝘗(G, 𝓂) (resp. 𝘗ᴅᴘ(G, 𝓂)< 𝘗(G, 𝓂)) holds for sufficiently large integers 𝓂. It is an interesting problem of finding graphs 𝘎 for which 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᴅᴘ(𝘎, 𝓂)< 𝘗(𝘎, 𝓂)) holds. Kaul and Mudrock showed that if 𝘎 has an even girth, then 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎) and Mudrock and Thomason recently proved that 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) holds for each graph 𝘎 which has a dominating vertex. We shall generalize their results in this article. For each edge e in 𝘎, let 𝓁(e)=∞ if e is a bridge of 𝘎, and let 𝓁(e) be the length of a shortest cycle in 𝘎 containing e otherwise. We first show that if 𝓁(e) is even for some edge e in 𝘎, then 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) holds for every graph G that contains a spanning tree T such that for each e∈E(G)∖E(T), 𝓁(e) is odd and e is contained in a cycle C of length ℓ(e) with the property that 𝓁(e′)< 𝓁(e) for each e′∈E(C)∖(E(T)∪{e}). Some open problems are proposed in this article.

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.

For any connected graph G, let P(G,m) and PDP (G,m) denote the chromatic polynomial and DP color function of G, respectively. It is known that PDP (G,m) ≤ P(G,m) holds for every positive integer m. Let DP≈ (resp. DP<) be the set of graphs G for which there exists an integer M such that PDP (G,m) = P(G,m) (resp. PDP (G,m) < P(G,m)) holds for all integers m ≥ M. Determining the sets DP≈ and DP< is an important open problem on the DP color function. For any edge set E0 of G, let ℓG(E0) be the size of a shortest cycle C in G such that |E(C) ∩ E0| is odd if such a cycle exists, and ℓG(E0) = ∞ otherwise. We denote ℓG(E0) as ℓG(e) if E0 = {e}.

In this paper, we prove that if G has a spanning tree T such that ℓG(e) is odd for each e ∈ E(G) \E(T), the edges in E(G) \E(T) can be labeled as e1, e2, . . . , eq with ℓG(ei) ≤ ℓG(ei+1) for all 1 ≤ i ≤ q−1 and each edge ei is contained in a cycle Ci of size ℓG(ei) with E(Ci) ⊆ E(T) ∪ {ej : 1 ≤ j ≤ i}, then G is a graph in DP≈. As a direct application, all plane near-triangulations and complete multipartite graphs with at least three partite sets belong to DP≈. We also show that if E∗ is a set of edges in G such that ℓG(E∗) is even and E∗ satisfies certain conditions, then G belongs to DP<. In particular, if ℓG(E∗) = 4, where E∗ is a set of edges between two disjoint vertex subsets of G, then G belongs to DP<. Both results extend known ones by Dong and Yang.

Let G be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the ow polynomial F(G; ) of G are integers if and only if G is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph G for which F(G; ) has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for G is positive if and only if F(G; ) does not have any real root in the interval (1; 2). We also prove that for any non-separable and 3-edge connected G, if G - e is also non-separable for each edge e in G and every 3-edge-cut of G consists of edges incident with some vertex of G, then all roots of P(G; ) are real if and only if either G 2 fL;Z3;K4g or F(G; ) contains at least 9 real roots in the interval (1; 2), where L is the graph with one vertex and one loop and Z3 is the graph with two vertices and three parallel edges joining these two vertices.

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.

For a given graph G, let P (G , A) be the chromatic polynomial of G, where A is considered to be a real number. In this paper, we study the bounds for P (G , A )/P (G , A — 1) and P (G , A )/P (G - x, A), where x is a vertex in G, A > n and n is the number of vertices of G.

The skewness of a graph 𝐺 is the minimum number of edges in 𝐺 whose removal results in a planar graph. It is a parameter that measures how nonplanar a graph is, and it also has important applications to VLSI design, but there are few results for skewness of graphs. In this paper, we first prove that the skewness is additive for the Zip product under certain conditions. We then present results on the lower bounds for the skewness of Cartesian products of graphs with trees and paths, respectively. Some exact values of the skewness for Cartesian products of complete graphs with trees, as well as of stars and wheels with paths are obtained by applying these lower bounds.