Graph Coloring Problems and Group Connectivity

1. Group connectivity. Let A be an abelian group and let iA(G) be the smallest positive integer m such that Lm(G) is A-connected. A path P of G is a normal divalent path if all internal vertices of P are of degree 2 in G and if |E(P)|= 2, then P is not in a 3-cycle of G. Let l(G) = max{lcub}m : G has a normal divalent path of length m{rcub}. We obtain the following result. (i) If |A| ≥ 4, then iA( G) ≤ l(G). (ii) If | A| ≥ 4, then iA(G) ≤ |V(G)| -- Delta(G). (iii) Suppose that |A| ≥ 4 and d = diam( G). If d ≤ |A| -- 1, then iA(G) ≤ d; and if d ≥ |A|, then iA(G) ≤ 2d -- |A| + 1. (iv) iZ 3 (G) ≤ l(G) + 2. All those bounds are best possible.;2. Modulo orientation. A mod (2p + 1)-orientation D is an orientation of G such that d +D(v) = d--D(v) (mod 2p + 1) for any vertex v ∈ V ( G). We prove that for any integer t ≥ 2, there exists a finite family F = F(p, t) of graphs that do not have a mod (2p + 1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. In particular, the graph family F(p, 2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.;3. Neighbor sum distinguishing total coloring. A proper total k-coloring &phis; of a graph G is a mapping from V(G) ∪ E(G) to {lcub}1,2, . . .,k{rcub} such that no adjacent or incident elements in V(G) ∪ E( G) receive the same color. Let m&phis;( v) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if m &phis;(u) ≠ m&phis;( v) for each edge uv ∈ E( G ). Let chitSigma(G) be the neighbor sum distinguishing total chromatic number of a graph G. Pilsniak and Wozniak conjectured that for any graph G, chitSigma( G) ≤ Delta(G) + 3. We show that if G is a graph with treewidth ℓ ≥ 3 and Delta(G) ≥ 2ℓ + 3, then chitSigma( G) + ℓ -- 1. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when ℓ = 3 and Delta ≥ 9, we show that Delta(G)+1 ≤ chit Sigma(G) ≤ Delta(G)+2 and characterize graphs with equalities.;4. Star edge coloring. A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Let ch\u27st(G) be the list star chromatic index of G: the minimum s such that for every s-list assignment L for the edges, G has a star edge coloring from L. By introducing a stronger coloring, we show with a very concise proof that the upper bound of the star chromatic index of trees also holds for list star chromatic index of trees, i.e. ch\u27st( T) ≤ [3Delta/2] for any tree T with maximum degree Delta. And then by applying some orientation technique we present two upper bounds for list star chromatic index of k-degenerate graphs

Progress on the adjacent vertex distinguishing edge colouring conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.Comment: v2: Revised following referees' comment

On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture

International audienceThis paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices

Coloring squares of graphs with mad constraints

A proper vertex $k$-coloring of a graph $G=(V,E)$ is an assignment $c:V\to \{1,2,\ldots,k\}$ of colors to the vertices of the graph such that no two adjacent vertices are associated with the same color. The square $G^2$ of a graph $G$ is the graph defined by $V(G)=V(G^2)$ and $uv \in E(G^2)$ if and only if the distance between $u$ and $v$ is at most two. We denote by $\chi(G^2)$ the chromatic number of $G^2$, which is the least integer $k$ such that a $k$-coloring of $G^2$ exists. By definition, at least $\Delta(G)+1$ colors are needed for this goal, where $\Delta(G)$ denotes the maximum degree of the graph $G$. In this paper, we prove that the square of every graph $G$ with $\text{mad}(G)<4$ and $\Delta(G) \geqslant 8$ is $(3\Delta(G)+1)$-choosable and even correspondence-colorable. Furthermore, we show a family of $2$-degenerate graphs $G$ with $\text{mad}(G)<4$, arbitrarily large maximum degree, and $\chi(G^2)\geqslant \frac{5\Delta(G)}{2}$, improving the result of Kim and Park.Comment: 14 pages, 4 figure