This report gather some notable results found in [SS12], giving also details on some foundation results presented in [CMN+07]. The results covered in Section 2 are answers given in [SS12] to questions of the article [CMN+07]. Graph parameters are studied a lot as they are a way to highlight a general structure from a graph. In that sense, the chromatic number is studied a lot as it shows the mu...

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree ∆ and with no isolated edges has an avd-coloring with at most ∆ + 300 colors, provided that ∆ > 1020. AMS Subject Classification: 05C15

Let χc(H) denote the circular chromatic number of a graph H. For graphs F and G, the circular chromatic Ramsey number Rχc(F,G) is the infimum of χc(H) over graphs H such that every red/blue edge-coloring of H contains a red copy of F or a blue copy of G. We characterize Rχc(F,G) in terms of a Ramsey problem for the families of homomorphic images of F and G. Letting zk = 3 − 2 −k, we prove that ...

Let us say a graph G has “tree-chromatic number” at most k if it admits a tree-decomposition (T, (Xt : t ∈ V (T ))) such that G[Xt] has chromatic number at most k for each t ∈ V (T ). This seems to be a new concept, and this paper is a collection of observations on the topic. In particular we show that there are graphs with tree-chromatic number two and with arbitrarily large chromatic number; ...

In this paper we define and study the distinguishing chromatic number, χD(G), of a graph G, building on the work of Albertson and Collins who studied the distinguishing number. We find χD(G) for various families of graphs and characterize those graphs with χD(G) = |V (G)|, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks’ Theorem for both the...

‎let $f$ be a proper $k$-coloring of a connected graph $g$ and‎ ‎$pi=(v_1,v_2,ldots,v_k)$ be an ordered partition of $v(g)$ into‎ ‎the resulting color classes‎. ‎for a vertex $v$ of $g$‎, ‎the color‎ ‎code of $v$ with respect to $pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_pi}(v)=(d(v,v_1),d(v,v_2),ldots,d(v,v_k))$‎, ‎where $d(v,v_i)=min{d(v,x):~xin v_i}‎, ‎1leq ileq k$‎. ‎if‎ ‎distinct...