A bcolouring of a graph g is a variant of proper kcolouring such. Also the relationship between dominator chromatic number and star chromatic number of the above graph families are obtained. I spent quite some time playing around with different colorings and incorrectly concluded the chromatic number was 4 because i could not at the time find one using 3 colors. Inapproximability results for both problems are given in 16. Characterization of forbidden subgraphs for bounded star. Actual colors have the disadvantage of being nite in number, which is. Prove that isomorphic graphs have the same chromatic number and the same chromatic polynomial. Isksn in a sense, xg corresponds to the best possible coloring of g, which may be better than the coloring corresponding to the ordinary chromatic number. Graph theory, bchromatic number, star chromatic number, achromatic number, equitable chromatic number, mathematics, qa1939, science, q publisher. The star chromatic number of a graph, a concept introduced by vince, is natural generalization of the chromatic number of a graph. The concept of the star chromatic number of a graph was introduced by vince a. A proper coloring of the vertices of a graph g is called a star coloring if the union of every two color classes induces a star forest.
In this paper, we survey results on this topic, concentrating on the relations among the circular chromatic number, the chromatic number and some other parameters of a graph. A graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. For example, the fact that a graph can be trianglefree. G, is the minimum number for which g has a proper kcolouring. Request pdf star chromatic number a generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the. Star list chromatic number of planar subcubic graphs. Toward a nordhausgaddum inequality for the number of dominating sets keough, lauren and shane, david, involve. G2 ianordered starcoloringof g is a partition of vg into independent sets v 1v k such that for i star chromatic numbers 555 3. The star chromatic number of a graph g is the least number of colours needed to colour the path on four vertices using three distinct colours. Since the publication of 11 additional articles 1, 2, 4, 9, 12 and have been written on the subject. A snark is a connected, bridgeless cubic graph with chromatic index 4. Definition a proper coloring of a graph is a function. G is said to be kchromatic if k is the least integer for which g has a kcoloring and. Finally, a proper vertex coloring of a graph is called a star coloring if the union of every two color classes induces a star forest.
Star chromatic number vince 1988 journal of graph theory wiley online library. Introduction the chromatic number xg of a graph is the least number of colors required for a proper vertex coloring of g. The chromatic number of a graph can be used in many realworld situations such as. Can you attach a new vertex to a graph to increase its chromatic number. Star chromatic number in light of theorem 2, define the star chromatic number xg as the least of the 2,chromatic numbers. Let g be a graph with vertex set vvg and edge set eeg. Star chromatic number in light of theorem 2, define the star chromatic number xg as the least of the 2, chromatic numbers.
The set of vertices with a specific colour is called a colour class. Dominator chromatic number of central, middle and total graphs of star and double star graph families are found in this paper. We then study the starchromatic number of an infinite family of graphs is constructed to show that for each. If two graphs are isomorphic, need they have the same chromatic number. The star chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. Chromatic index is analogous to chromatic number but deals with edge colorings. Yet, as far as we know, there has been no attempt in trying to characterize the forbidden structures to obtain bounded acyclic chromatic number or star chromatic number. Pdf matrices of graphs and designs with emphasis on their. A star colouring of a graph gis a proper vertex colouring in which every path on four vertices uses at least three distinct colours. Blue purple nebula star field travel hd longest motion background aa vfx duration.
In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. Star chromatic numbers of some planar graphs, journal of. Chapter 3 bchromatic number of graphs and transformation. Pdf total chromatic number of double star graph families. Graph coloring and chromatic numbers brilliant math.
A star coloring 3 of a graph is a proper vertex coloring in which every pathon four vertices uses at least three distinct. Pdf on jan 1, 2018, g jayaraman and others published total chromatic number of double star graph families find, read and cite all the research you need on researchgate. In this paper we initiate the study of the star chromatic number and star defining number. Departement of mathematics, zhejiang normal university, jinhua 321004, p. A note on star chromatic number of graphs 843 adjacent i. A kcoloring of a graph is a vertex coloring of g that uses kcolors.
If g is a connected graph with n vertices, xg min xkg. Is there an efficient way for finding the chromatic number. The pentagon is an odd cycle, which we showed was not bipartite. The star chromatic numbers of some planar graphs derived from. Any such minimum partition is called a chromatic partition of vg. The circular chromatic number of graph g was defined by vince 20, where he called is the star chromatic number, to be the infimum over all rational numbers n k such that there exists a map from. It is impossible to color the graph with 2 colors, so the graph has chromatic number 3. Sep 01, 2018 as it turns out, the star dichromatic number is a lower bound for the circular dichromatic number of bokal et al. The path partition conjecture is true and its validity.
On an exam, i was given the peterson graph and asked to find the chromatic number and a vertex coloring for it. A study on star chromatic number of some special classes. Some theorems concerning the star chromatic number of a graph bing zhou department of mathematics, trent university, peterborough, ontario, canada k9j 7b8 received june 10, 1994 we investigate the notion of the star chromatic number of a graph in conjunction with various other graph parameters, among them, clique number, girth, and independence. If we assign colors to nv and nu such that the assignment colors to nv are different with the assignment colors to nu, then this colorings is said to be vertex star colorings. Chromatic index is analogous to chromatic number but deals with. G of a graph g is the minimum number of independent subsets that partition the vertex set of g. If number of vertices having degree 1 is n1 and number of vertex having degree n1 is 1 then our graph should be a star graph otherwise it should be not. If g is l star colorable for any list assignment l with lv.
Star chromatic numbers and products of graphs wiley online library. Multichromatic numbers, star chromatic numbers and kneser. Gis equivalent with acyclicstar vertex coloring of the linegraph lg. Some theorems concerning the star chromatic number of a graph. Just traverse whole matrix and record the number of vertices having degree 1 and degree n1. The star chromatic number is the minimum number of colours needed to star colour g 1and is denoted. The star chromatic numbers of some planar graphs derived. By erdos 1959, there is a graph with large girth and large chromatic number. One generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the twocolor induced subgraphs are forests.
We introduce the circlechromatic number of a graph and prove that for any graph g. The notion of the star chromatic number of a graph is a generalization of the chromatic number. The chromatic number of a graph is the minimum number of colors needed to produce a proper coloring of the graph. We then study the star chromatic number of an infinite family of graphs is constructed to show that for each. We then study the starchromatic number of colorcritical graphs.
Golomb graph wikiwand circular chromatic numbers and fractional chromatic numbers of. The maximum acyclic chromatic number on g is 43log and o alon, mcdiarmid, and reed 2. X2 6 since the occurrence of each event rx1 depends only on the color of the vertices in x1, g. As a consequence of the path partition theorem, we have obtained an upper bound for the star chromatic number. A note on the star chromatic number wiley online library. What is chromatic number definition and meaning math. Graph theory, b chromatic number, star chromatic number, achromatic number, equitable chromatic number, mathematics, qa1939, science, q publisher.
In this paper chromatic number of sunlet and bistar families of graphs are found. Star coloring of sparse graphs virginia commonwealth university. The least number of colors require to color the vertices of a graph so that the adjacent vertices do not have the same color is called as the chromatic number. Pdf we investigate the relation between the starchromatic number. Pdf b chromatic number of line graph of wheel graph. The star chromatic number of g is the least number of colors needed to star color g. V g n such that every two adjacent vertices receive different col ours. Key words central graph, middle graph, total graph and dominator coloring. A proper kcolouring of a graph g v g, e g is a mapping f. Dominator coloring on star and double star graph families. Thus, if the class of free graphs has bounded chromatic number then is a forest.
Gnx chromatic number of some cycle related graph has examined by vaidhya and shukla and also they established the b chromatic number of wheel related graph11. The graph g is l star colorable if for a given list assignment l there is a star coloring. How does deleting an edge from a complete graph e ect its chromatic number. This paper calculates the star chromatic numbers of three infinite families of planar graphs. Star chromatic index zden ek dvo r ak bojan mohar yz robert s amal x abstract the star chromatic index. Abstract 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. Star chromatic numbers and products of graphs zhu 1992.
Researchers have obtained a few positive algorithmic results for these problems on graphs for which the. More precisely, the first family of planar graphs has star chromatic numbers. We say a graph g is star extremal if its star chromatic number is. Multichromatic numbers, star chromatic numbers and kneser graphs johnson, a holroyd, f. We direct the readers to the wellknown graph coloring book 10 of jensen and toft for more information regarding various coloring parameters. G of a graph g g g is the minimal number of colors for which such an.
Some theorems concerning the star chromatic number of a. We point out an alternate definition of the star chromatic number, which sheds new light on the relation of the star chromatic number and the ordinary chromatic number. Star 4 chromatic unit distance graph star 4 chromatic unit distance graph by. A way to consider this is using the principle of inclusionexclusion. A proper vertex colouring of a graph g is called star colouring 10, if every path of g on four vertices is not 2 coloured. If one considers the class of graphs g of maximum degree at most, brooks theorem shows that the usual chromatic number is o. Graph theory 12 1988, 551559, which is a natural generalization of the chromatic number of a graph. Induced subgraphs of graphs with large chromatic number. The star chromatic number of g is the smallest integer k for which g admits a star coloring with k colors. Pdf in this paper, we investigate the b chromatic number of central graph, middle graph and total graph of star graph, denoted by ck1,n, mk1,n and. We conclude with a discussion of the case of planar digraphs and point out some open problems. Star chromatic number star chromatic number vince, a. A greedy colouring relative to a vertex orderingv1 0 and eachm. Circular chromatic number for iterated mycielski graphs.
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