Locally Identifying Coloring of Graphs

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known \((\varDelta ,D)\) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. For instance, we provide an asymptotically optimal construction for \((n,n+k)\) cubic graphs - its forwarding index is \(\sim \frac{n^2}{3k} \log _2(k)\). Our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e. least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected?

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

Exploring the complexity boundary between coloring and list-coloring

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard.

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

An adjacent vertex distinguishing edge coloring of a graph $$G$$ is a proper edge coloring of $$G$$ such that any pair of adjacent vertices admit different sets of colors. The minimum number of colors required for such a coloring of $$G$$ is denoted by $$\chi ^{\prime }_{a}(G)$$ . In this paper, we prove that if $$G$$ is a planar graph with girth at least 5 and $$G$$ is not a 5-cycle, then $$\chi ^{\prime }_{a}(G)\le \Delta +2$$ , where $$\Delta $$ is the maximum degree of $$G$$ . This confirms partially a conjecture in Zhang et al. (Appl Math Lett 15:623---626, 2002).

Linear coloring of planar graphs with large girth

Odd graph and its applications to the strong edge coloring

1-planar graphs are odd 13-colorable

in this paper we introduce a new concept of color Laplacian energy LEc (G). It depends on the underlying graph G and colors of the vertices.We compute colors of the vertices. We compute color Laplacian spectrum and color Laplacian energies of families of graph with minimum number of colors. We also obtain some bounds of color Laplacian energy. The color Laplacian energy for colored compliment of few families of graphs are also obtained.

Clustering and domination in perfect graphs

The coloring game on planar graphs with large girth, by a result on sparse cactuses

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud, and Wang conjectured that every planar graph of maximum degree Δ ≥ 3 admits an injective colouring with at most ⌊3Δ/2⌋ colours. This was later disproved by Lužar and Škrekovski for certain small and even values of Δ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lužar and Škrekovski’s conjecture and extend their results by presenting an infinite family of 3-connected planar graphs for each Δ (except for 4) attaining their bound, whereas they only gave finitely many examples for each Δ. Hence, together with another infinite family of maximum degree 4, we provide infinitely many counterexamples to the conjecture by Chen et al. for each Δ if 4 ≤ Δ ≤ 7 and every even Δ ≥ 8. We provide similar evidence for analogous conjectures by La and Štorgel and Lužar, Škrekovski, and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of 3-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees Δ ≥ 3.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

A dominator coloring of a graph $G$ is a proper coloring in which every vertex of $G$ dominates at least one color class. The dominator chromatic number of $G, \chi_d(G)$, is defined by the minimum number of colors needed in a dominator coloring of $G$. A total dominator coloring of $G$ is a proper coloring of $G$ with the extra property that every vertex in $G$ properly dominates a color class. The total dominator chromatic number of $G, \chi_{t d}(G)$, is the minimum number of colors needed in a total dominator coloring of $G$. In this paper, we obtain dominator and total dominator chromatic number of Mongolian tent graphs and fire cracker graphs.