The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges, 1. THE CHROMATIC SUM A proper coloring of the vertices of the graph G must assign different colors to adjacent vertices. The chroma...

In the Domination Game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set A∗ dominates the whole graph. The Dominator plays to minimise the size of the set A∗ while the Staller plays to maximise it. A graph is D-trivial if when the Dominator plays first and both p...

Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Conjecture. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Several families of snarks have had their total...

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

Previous studies on augmented reality (AR) book learning have not provided an in-depth examination of the learning process, especially the interaction involved in child–parent shared book reading. Choosing an AR picture book to introduce its artistic work, this study aimed to explore how children and parents read the book through a series of analyses of behavioral patterns and cognitive attainm...

Reed conjectured that for every > 0 and ∆ there exists g such that the fractional total chromatic number of a graph with maximum degree ∆ and girth at least g is at most ∆ + 1 + . We prove the conjecture for ∆ = 3 and for even ∆ ≥ 4 in the following stronger form: For each of these values of ∆, there exists g such that the fractional total chromatic number of any graph with maximum degree ∆ and...

‎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...

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