A study of the total coloring of graphs.

The area of total coloring is a more recent and less studied area than vertex and edge coloring, but recently, some attention has been given to the Total Coloring Conjecture, which states that each graph\u27s total chromatic number xT is no greater than its maximum degree plus two. In this dissertation, it is proved that the conjecture is satisfied by those planar graphs in which no vertex of degree 5 or 6 1ies on more than three 3-cycles. The total independence number aT is found for some families of graphs, and a relationship between that parameter and the size of a graph\u27s minimum maximal matching is discussed. For colorings with natural numbers, the total chromatic sum ST is introduced, as is total strength (oT of a graph. Tools are developed for proving that a total coloring has minimum sum, and this sum is found for some graphs including paths, cycles, complete graphs, complete bipartite graphs, full binary trees, and some hypercubes. A family of graphs is found for which no optimal total coloring maximizes the smallest color class. Lastly, the relationship between a graph\u27s total chromatic number and its total strength is explored, and some graphs are found that require more than their total chromatic number of colors to obtain a minimum sum

On Minimizing the Maximum Color for the 1-2-3 Conjecture

The 1-2-3 Conjecture asserts that, for every connected graph different from K2 , its edges can be labeled with 1,2,3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graphs so that some pairs of their elements are distinguishable relatively to some parameter computed from the labeling. In this work, we investigate the consequences of labeling graphs as in the 1-2-3 Conjecture when it is further required to make the maximum resulting color as small as possible. In some sense, we aim at producing a number of colors that is as close as possible to the chromatic number of the graph. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permits to reduce the maximum color

On Minimizing the Maximum Color for the 1-2-3 Conjecture

International audienceThe 1-2-3 Conjecture asserts that, for every connected graph different from K2 , its edges can be labeled with 1,2,3 so that, when coloring each vertex with the sum of its incident labels, no two adjacent vertices get the same color. This conjecture takes place in the more general context of distinguishing labelings, where the goal is to label graphs so that some pairs of their elements are distinguishable relatively to some parameter computed from the labeling. In this work, we investigate the consequences of labeling graphs as in the 1-2-3 Conjecture when it is further required to make the maximum resulting color as small as possible. In some sense, we aim at producing a number of colors that is as close as possible to the chromatic number of the graph. We first investigate the hardness of determining the minimum maximum color by a labeling for a given graph, which we show is NP-complete in the class of bipartite graphs but polynomial-time solvable in the class of graphs with bounded treewidth. We then provide bounds on the minimum maximum color that can be generated both in the general context, and for particular classes of graphs. Finally, we study how using larger labels permits to reduce the maximum color

Coloring Sums of Extensions of Certain Graphs

Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $\chi(G).$ In this paper the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum are introduced. The extended graph $G^x$ of a graph $G$ was recently introduced for certain regular graphs. We further the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum to extended paths and cycles. The paper concludes with \emph{patterned structured} graphs.Comment: 12 page

Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems

Vertex-Coloring 2-Edge-Weighting of Graphs

A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined on an abelian group. It was proved that every 3-colorable graph admits a vertex-coloring $3$-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In particular, we show that 3-connected bipartite graphs admit vertex-coloring 2-edge-weighting