Some problems on induced subgraphs

Analyzing chromatic number in coloring problem is a tough topic in graph analysis. We focus on the basic theory for a particular type of chromatic number. This will give us insights on the basic topological structure guiding lots of networks in the coming trend of big data era. An adjacent vertex-distinguishing total k-coloring is a proper total k-coloring of a graph G such that for any two adjacent vertices, the set of colors appearing on the vertex and its incident edges are different. The smallest k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number, and denoted by ?at(G). It has been proved that if the graph G satisfies ?(G)=3, then ?at(G)= 6. However, it is very difficult to determine whether ?at(G)= 5. In this paper, we focus on a special class of 3-regular graphs, the generalized Petersen graphs P(n, k), and show that ?at(P(n, k)) = 5, which improves the bound ?at (P(n, k))= 6.

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let $$i_k = i_k(G)$$ik=ik(G) be the number of independent sets of cardinality k of G. The independence polynomial $$I(G, x)=\sum _{k\geqslant 0}i_k(G)x^k$$I(G,x)=źkź0ik(G)xk defined first by Gutman and Harary has been the focus of considerable research recently, whereas $$i(G)=I(G, 1)$$i(G)=I(G,1) is called the Merrifield---Simmons index of G. In this paper, we first proved that among all trees of order n, the kth coefficient $$i_k$$ik is smallest when the tree is a path, and is largest for star. Moreover, the graph among all trees of order n with diameter at least d whose all coefficients of I(G, x) are largest is identified. Then we identify the graphs among the n-vertex unicyclic graphs (resp. n-vertex connected graphs with clique number $$\omega $$ź) which simultaneously minimize all coefficients of I(G, x), whereas the opposite problems of simultaneously maximizing all coefficients of I(G, x) among these two classes of graphs are also solved respectively. At last we characterize the graph among all the n-vertex connected graph with chromatic number $$\chi $$ź (resp. vertex connectivity $$\kappa $$ź) which simultaneously minimize all coefficients of I(G, x). Our results may deduce some known results on Merrifield---Simmons index of graphs.

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the \emph{nested complexity length}, which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large `half-graphs' do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large complete multipartite graphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

The structural information contentIg(X) of a graphX was treated in detail in three previous papers (Mowshowitz 1968a, 1968b, 1968c). Those investigations ofIg point up the desirability of defining and examining other entropy-like measures on graphs. To this end the chromatic information contentIc(X)of a graphX is defined as the minimum entropy over all finite probability schemes constructed from chromatic decompositions having rank equal to the chromatic number ofX. Graph-theoretic results concerning chromatic number are used to establish basic properties ofIc on arbitrary graphs. Moreover, the behavior ofIc on certain special classes of graphs is examined. The peculiar structural characteristics of a graph on which the respective behaviors of the entropy-like measuresIc andIg depend are also discussed.

Circular chromatic numbers of Mycielski's graphs

S-functions are mappings from the class of finite graphs into the set of integers, such that certain formal conditions are fulfilled which are shared by the chromatic number, the vertex-connectivity, and the homomorphism-degree. The S-functions form a complete lattice (with respect to their natural partial order). The classes of graphs with values <n under some S-function are studied from a general point of view, and uncountably many S-functions are constructed. Further for every n≥5 a non-trivial base-element of (see K. WAGNER [7]) is constructed.

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

This chapter describes the history of graphs and introduces the main definitions of graphs. It discusses some particular classes of graphs that we can meet in philosophy and provides some examples of them. The chapter presents a list of well-known graphs that we can put together according to their name. A graph generally has different kinds of drawings, but all of these are isomorphic. So, a graph is in fact a class of graphs generally defined up to an isomorphism. Even when graphs are non-separable, they always admit subgraphs. Informally speaking, the reconstruction conjecture says that graphs are determined uniquely by their subgraphs. The conjecture has been verified for a number of infinite classes of graphs: regular graphs, trees, disconnected graphs, unit interval graphs, separable graphs without end vertices, maximal planar graphs, maximal outerplanar graphs, outerplanar graphs and critical blocks.

In 2013, C. Godsil introduced the quantum version of Hedetniemi’s conjecture. To solve this conjecture, we introduce the concept of quantum multiplicative graph and obtain a necessary and sufficient condition that a graph is quantum multiplicative. We also examine that the quantum chromatic number of the categorical product of a class of graphs and construct a class of graphs exhibiting separations between clique number and quantum clique number in this paper.

The quantum chromatic number of a graph $G$ is sandwiched between its\nchromatic number and its clique number, which are well known NP-hard\nquantities. We restrict our attention to the rank-1 quantum chromatic number\n$\\chi_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is\ndefined under stronger constraints. We study its relation with the chromatic\nnumber $\\chi(G)$ and the minimum dimension of orthogonal representations\n$\\xi(G)$. It is known that $\\xi(G) \\leq \\chi_q^{(1)}(G) \\leq \\chi(G)$. We\nanswer three open questions about these relations: we give a necessary and\nsufficient condition to have $\\xi(G) = \\chi_q^{(1)}(G)$, we exhibit a class of\ngraphs such that $\\xi(G) &lt; \\chi_q^{(1)}(G)$, and we give a necessary and\nsufficient condition to have $\\chi_q^{(1)}(G) &lt; \\chi(G)$. Our main tools are\nKochen-Specker sets, collections of vectors with a traditionally important role\nin the study of noncontextuality of physical theories, and more recently in the\nquantification of quantum zero-error capacities. Finally, as a corollary of our\nresults and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum\nchromatic number, we give a family of Kochen-Specker sets of growing dimension.\n

Beck’s conjecture and multiplicative lattices

New bounds for the max-k-cut and chromatic number of a graph

Asymptotic Clique Covering Ratios of Distance Graphs