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Intuitionistic fuzzy graphs (IFGs) are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real‐world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring (IFVOC) and analyzed the alpha‐strong (α˜str), beta‐strong (β˜str), and gamma‐strong (γ˜str) vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of α˜str vertices, the chromatic number, and the weight of the graph's minimum spanning tree. We systematically evaluate different types of IFG products, such as conormal, modular, residue, and maximal, and consider their implications for fuzzy graph structure and connectivity. We have investigated new metrics to measure the presence and importance of α˜str vertices in the graph, assessing both their frequency and overall impact. Additionally, we have investigated the impact of product operations on the significance of α˜str nodes and the resultant minimum spanning tree, providing insights into the overall robustness and efficiency of each product. We have explored how variations in product operations impact the distribution of α˜str vertices and the characteristics of the minimum spanning tree and chromatic number, elucidating the trade‐offs between different product options. We have modeled the product graph as a network to represent complex systems composed of interconnected entities. The outcomes of this research contribute to the advancement of IFG theory and its use in various fields, such as network design, strategic planning, and system analysis. [ABSTRACT FROM AUTHOR]

Objectives: The aim of this article is to enhance the understanding of the computation of chromatic polynomials and chromatic index polynomials, and to facilitate their practical use in various fields by demonstrating and supporting the proposed algorithms. Results: The theorems for computing the chromatic polynomial and chromatic index polynomial are thoroughly demonstrated and validated in this article. Additionally, the algorithms used in these computations are introduced and explained.

A graph G is chromatically unique if its chromatic polynomial completely determines the graph. An n-spoked wheel, Wn, is shown to be chromatically unique when n ≥ 4 is even [S.-J. Xu and N.-Z. Li, The chromaticity of wheels, Discrete Math. 51 (1984) 207-212]. When n is odd, this problem is still open for n ≥ 15 since 1984, although it was shown by different researchers that the answer is no for n = 5, 7, yes for n = 3, 9, 11, 13, and unknown for other odd n. We use the beta invariant of matroids to prove that if M is a 3-connected matroid such that |E(M)| = |E(Wn)| and β(M) = β(M(Wn)), where β(M) is the beta invariant of M, then M ∼= M(Wn). As a consequence, if G is a 3-connected graph such that the chromatic (or flow) polynomial of G equals to the chromatic (or flow) polynomial of a wheel, then G is isomorphic to the wheel. The examples for n = 3, 5 show that the 3-connectedness condition may not be dropped. We also give a splitting formula for computing the beta invariants of general parallel connection of two matroids as well as the 3-sum of two binary matroids. This generalizes the corresponding result of [T.H. Brylawski, A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1-22]. [ABSTRACT FROM AUTHOR]

In 1997, Sierpinski graphs, S (n , k) , were obtained by Klavzar and Milutinovic. The graph S (1 , k) represents the complete graph K k and S (n , 3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by S i e (Λ , t) , already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph S i e (Λ , t) . In the second step, we extract some degree-based indices of a generalized Sierpinski graph S i e (Λ , t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph S i e (Λ , t) by using the Randić index. [ABSTRACT FROM AUTHOR]

Graph Theory is a discipline of mathematics with numerous outstanding issues and applications in a variety of sectors of mathematics and science. The chromatic polynomial is a type of polynomial that has useful and attractive qualities. Ehrhart's polynomials and chromatic analysis are two essential techniques for graph analysis. They both provide insight into the graph's structure but in different ways. The relationship between chromatic and Ehrhart polynomials is an area of active research that has implications for graph theory, combinatorial, and other fields. By understanding the relationship between these two polynomials, one can better understand the structure of graphs and how they interact with each other. This can help us to solve complex problems in our lives more efficiently and effectively. This work gives the relationship between these two essential polynomials and the proof of theorems and an application related to these works was discussed, which is the model Physical Cell ID (PCID).

We introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group 𝕲 acting on a graph G and a 𝕲-set X, a proper X-coloring is a function with no monochromatic edge orbit. The set of proper colorings is a 𝕲-set which induces a polynomial function from the Burnside ring of 𝕲 to itself. In this paper, we study many properties of the Burnside chromatic polynomial, answering some questions of Zaslavsky.

Objective: In the field of graph theory, maple is a technical computation form that is used for solving problems. In this article, we apply maple to find the strong fuzzy chromatic polynomial of fuzzy graphs and related. Moreover, we apply maple to obtain strong fuzzy chromatic numbers of fuzzy graphs using their strong fuzzy chromatic polynomials. Results: The strong fuzzy chromatic polynomials for fuzzy graphs, strong fuzzy graphs and complete fuzzy graphs are determined using maple. Furthermore, the strong fuzzy chromatic numbers for the fuzzy graphs are obtained. [ABSTRACT FROM AUTHOR]

The Tutte polynomial is an isomorphism invariant of graphs that generalizes the chromatic and the flow polynomials. This two-variable polynomial with integral coefficients is known to carry important information about the properties of the graph. It has been used to prove long-standing conjectures in knot theory. Furthermore, it is related to the Potts and Ising models in statistical physics. The purpose of this paper is to study the interaction between the Tutte polynomial and graph symmetries. More precisely, we prove that if the automorphism group of the graph G contains an element of prime order p, then the coefficients of the Tutte polynomial of G satisfy certain necessary conditions. [ABSTRACT FROM AUTHOR]

The concept of graph colouring has become a very active field of research that enhances many practical applications and theoretical challenges. An accurate (vertex) colouring through the condition that each vertex is whichever unaccompanied in its colour class or adjoining to all vertices of at least one colour class is known as dominator colouring. A graph is dominator colouring is a suitable colouring in which at least one vertex dominates each colour class. Many difficult and global issues are solved using fuzzy graph colouring approaches. In this paper, we introduce a new Sugeno-Type Fuzzy graph of groups, which is found using several fuzzy graph operations such as dominant path-colouring number, cycle, union, join, and products. A number that is representative based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths is the minimum number of shared edges based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths. The sugeno dominant path-colouring number is the minimum Sugeno number of shared edges among the Sugeno cardinality of all sets of shared edges, which allows for various techniques. These results are used in studying various and recently introduced chromatic numbers of graphs. [ABSTRACT FROM AUTHOR]

The article describes a new thought on the chromatic polynomial of an intuitionistic fuzzy graph which is illustrated based on α , β -level graphs. Besides, the alpha-beta fundamental set of an intuitionistic fuzzy graph is also defined with a vivid description. In addition to that, some characterizations of the chromatic polynomial of an intuitionistic fuzzy graph are specified as well verified. Furthermore, some untouched properties of the α , β -level graph are also projected and proved. [ABSTRACT FROM AUTHOR]

The b-chromatic number b(G) of a graph G is the maximum k for which G has a proper vertex coloring using k colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we have mainly investigated on the b-chromatic number of the Mycielskian of regular graphs. In particular, we have obtained the exact value of the b-chromatic number of the Mycielskian of some classes of graphs. This includes a few families of regular graphs, graphs with b(G) = 2 and split graphs. In addition, we have found bounds for the b-chromatic number of the Mycielskian of some more families of regular graphs in terms of the b-chromatic number of their original graphs. Also we have found b-chromatic number of the generalized Mycielskian of some regular graphs. [ABSTRACT FROM AUTHOR]

For $m \geq 3$ and $n \geq 1$, the $m$-cycle \textit{book graph} $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.

We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number χ*(D) of a digraph D, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are "easier" or "harder" to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced in [G. Wang, S. Zhou, G. Liu and J. Wu, Circular vertex arboricity, J. Discrete Appl. Math. 159 (2011) 1231{1238] and resembles the concept of the star chromatic number of graphs introduced by Vince in [15] in the framework of digraph colouring. After presenting basic properties of the new quantity, including range, simple classes of digraphs, general inequalities and its relation to integer counterparts as well as other concepts of fractional colouring, we compare our notion with the notion of circular colourings for digraphs introduced in [D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004) 227{224] and point out similarities as well as differences in certain situations. As it turns out, the star dichromatic number shares all positive characteristics with the circular dichromatic number of Bokal et al., but has the advantage that it depends on the strong components of the digraph only, while the addition of a dominating source raises the circular dichromatic number to the ceiling. We conclude with a discussion of the case of planar digraphs and point out some open problems. [ABSTRACT FROM AUTHOR]

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the r-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function. [ABSTRACT FROM AUTHOR]

The connection between the matching polynomial and the chromatic polynomial for triangle-free graphs was revealed in the work of Farrell and Whitehead. We extend this result to all graph by mirroring the corresponding result of Godsil and Gutman for the acyclic polynomial and the characteristic polynomial. We also reintroduce the clique polynomial of Farrell as an evaluation of the U-polynomial of Noble and Welsh, which also happens to contain the matching and chromatic polynomials.

The search for counterexamples to the Four Color Conjecture originated snarks, a very special class of cubic graphs. In this paper, we consider the equitable total coloring of snarks. A total coloring is equitable if the number of elements colored with each color differs by at most one, and the least integer for which a graph has such a coloring is called its equitable total chromatic number. In 2002, Wang conjectured that the equitable total chromatic number of a graph is at most ∆ + 2, and this was proved for cubic graphs. Therefore, the equitable total chromatic number of a cubic graph is either 4 or 5. We provide evidence to a negative answer to the question proposed in 2016 about the existence of a Type 1 cubic graph with girth greater than 4 and equitable total chromatic number 5, by determining equitable 4-total colorings for every member of three infinite families of snarks with girth 5. [ABSTRACT FROM AUTHOR]

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either ∆(G) + 1 or ∆(G) + 2, where ∆(G) is the maximum degree of G. We consider the direct product of complete graphs K m × K n. It is known that if at least one of the numbers m or n is even, then K m × K n has total chromatic number equal to ∆(K m × K n) + 1, except when m = n = 2. We prove that the graph K m × K n has total chromatic number equal to ∆(K m × K n) + 1 when both m and n are odd numbers, ensuring in this way that all graphs K m × K n have total chromatic number equal to ∆ (K m × K n) + 1, except when m = n = 2. [ABSTRACT FROM AUTHOR]

Herein, we developed and analyzed a new fractal-fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton's solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature. [ABSTRACT FROM AUTHOR]

For a positive integer k ≥ 1, a graph G with vertex set V is said to be k-packing colorable if there exists a mapping f : V → {1, 2, ..., k} such that any two distinct vertices x and y with the same color f(x) = f(y) are at distance at least f(x) + 1. The packing chromatic number of a graph G, denoted by χρ(G), is the smallest integer k such that G is k-packing colorable. In this work, we study both independence and packing colorings in the m-generalized Mycielskian of a graph G, denoted μm(G). We first give an explicit formula for α (μm(G)) when m is odd and bounds when m is even. We then use these results to give exact values of α(μm(Kn)) for any m and n. Next, we give bounds on the packing chromatic number, χρ, of μm(G). We also prove the existence of large planar graphs whose packing chromatic number is 4. The rest of the paper is focused on packing chromatic numbers of the Mycielskian of paths and cycles. [ABSTRACT FROM AUTHOR]

It is surprising that despite so much literature on the four-color theorem, no one ever computed the number of ways to color an actual map, such as that of Canada, with four colors. For the curiosity of the novice, in this project, we did it for a far more challenging map of the USA. [ABSTRACT FROM AUTHOR]

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a contraction-extraction process. This gives Hopf-algebraic proofs of Rota's result on the signs of coefficients of chromatic polynomials and of Stanley's interpretation of the values at negative integers of chromatic polynomials. We also consider chromatic symmetric functions and their noncommutative versions. [ABSTRACT FROM AUTHOR]

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 = t = d (max T + 1): d | t ⊙ t/d ⊙ T}. We show that espd⊙T (G) = d espT (G) - r, where r, 0 = r = d - 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙T{0}(G) = ⊙d(c(G) - 1), where c(G) is the circular chromatic number of G. This result allows us to formulate several interesting conclusions that include a new formula for the circular chromatic number c(G) = 1 + inf espd⊙{0}(G)/d: d = 1 and a proof that the formula for the T-edge span of powers of cycles, stated as conjecture in [Y. Zhao, W. He and R. Cao, The edge span of T-coloring on graph Cd n, Appl. Math. Lett. 19 (2006) 647-651], is true. [ABSTRACT FROM AUTHOR]

A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper rainbow connection number of G, denoted by prc(G), is the minimum number of colors needed to properly color the edges of G so that G is rainbow connected. Similarly, the proper strong rainbow connection number of G, denoted by psrc(G), is the minimum number of colors needed to properly color the edges of G such that for any two distinct vertices of G, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if G = Kp1 · · · Kpn, where n = 1, and p1, . . ., pn > 1 are integers, then prc(G) = psrc(G) = X'(G), where X'(G) denotes the chromatic index of G. Finally, we investigate some sufficient conditions for a graph G to satisfy prc(G) = rc(G), and make some slightly positive progress by using a relation between rc(G) and the girth of the graph. [ABSTRACT FROM AUTHOR]

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph G, is a proper edge coloring of G such that each color class is dominated by at least one edge of G. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by X'0dom(G). We obtain some properties of X'0dom(G) and compute it for specific graphs. Also examine the effects on X'0dom(G), when G is modified by operations on vertex and edge of G. Finally, we consider the k-subdivision of G and study the dominated edge chromatic number of these kind of graphs. [ABSTRACT FROM AUTHOR]

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2. [ABSTRACT FROM AUTHOR]

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that/c(N(v))l > min (r, d(v)}, for each v £ V (G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r--dynamic chromatic number of the middle, central and line graphs of the gear graph. [ABSTRACT FROM AUTHOR]

A star edge coloring of a graph G is a proper edge coloring such that there is no bicolored path or cycle of length four. The minimum number of colors needed for a graph G to admit a star edge coloring is called the star chromatic index and it is denoted by χ′s(G). In this paper, we consider graphs of maximum degree Δ ≥ 4 and show that if the maximum average degree of a graph is less than 14/5 then χ′s(G) ≤ 2Δ + 1. [ABSTRACT FROM AUTHOR]

Objective. To develop a series of polynomial models to track the growth and trend of infection and death curve for COVID-19 in Colombia. Methods. The infected and daily deaths from COVID-19 between March 6 to April 10, 2021, were used. For its prediction analysis, we use polynomial functions in Excel. Results. Of the six polynomial functions evaluated, the polynomial with the highest level of determination is that of degree 6 according to the adjusted R.. Predictions were made taking into account the accumulated polynomial functions of confirmed infected and deceased. Conclusions. Easy-to-build Excel models such as polynomial functions are effective for monitoring public health events, facilitating timely decision-making. [ABSTRACT FROM AUTHOR]

Let G be a simple graph. The dominated coloring of G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of G is called the dominated chromatic number of G, denoted by dom(G). Stability (bondage number) of dominated chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the dominated chromatic number of G. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs. [ABSTRACT FROM AUTHOR]

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6] , the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2] , we proved that every cycle has this property. Here we give a common generalization. Say a "banana" is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned. [ABSTRACT FROM AUTHOR]

For a non-decreasing sequence of positive integers S = (s 1 , s 2 , ...) , the S -packing chromatic number χ S (G) of G is the smallest integer k such that the vertex set of G can be partitioned into sets X i , i ∈ k ] , where vertices in X i are pairwise at distance greater than s i. In this paper we introduce S -packing chromatic vertex-critical graphs, χ S -critical for short, as the graphs in which χ S (G − u) < χ S (G) for every u ∈ V (G). This extends the earlier concept of the packing chromatic vertex-critical graphs. We show that if G is χ S -critical, then the set { χ S (G) − χ S (G − u) : u ∈ V (G) } can be almost arbitrary. If G is χ S -critical and χ S (G) = k (k ∈ N), then G is called k - χ S -critical. We characterize 3- χ S -critical graphs and partially characterize 4- χ S -critical graphs when s 1 > 1. We also deal with k - χ S -criticality of trees and caterpillars. [ABSTRACT FROM AUTHOR]

It is proved that every connected graph G on n vertices with χ (G) ≥ 4 has at most k (k − 1) n − 3 (k − 2) (k − 3) k -colourings for every k ≥ 4. Equality holds for some (and then for every) k if and only if the graph is formed from K 4 by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29]. Proof methods may be of independent interest. In particular, one of our auxiliary results about list-chromatic polynomials solves a generalized version of a recent conjecture of Brown, Erey, and Li. [ABSTRACT FROM AUTHOR]

Fibonacci cubes are defined as subgraphs of hypercubes, where the vertices are those without two consecutive 1's in their binary string representation. k -Fibonacci cubes are in turn special subgraphs of Fibonacci cubes obtained by eliminating certain edges. This elimination is carried out at the step analogous to where the fundamental recursion is used to construct Fibonacci cubes themselves from the two previous cubes by link edges. In this work, we calculate the vertex chromatic polynomial of k -Fibonacci cubes for k = 1, 2. We also determine the domination number and the total domination number of k -Fibonacci cubes for n, k ≤ 12 by using an integer programming formulation. [ABSTRACT FROM AUTHOR]

Chromatic polynomials of graphs have been studied extensively for around one century. The concept of chromatic polynomial of a hypergraph is a natural extension of chromatic polynomial of a graph. It also has been studied for more than 30 years. This short article will focus on introducing some important open problems on chromatic polynomials of hypergraphs. [ABSTRACT FROM AUTHOR]

Kingdomino TM is a board game designed by Bruno Cathala and edited by Blue Orange since 2016. The goal is to place 2 × 1 dominoes on a grid layout, and get a better score than other players. Each 1 × 1 domino cell has a color that must match at least one adjacent cell, and an integer number of crowns (possibly none) used to compute the score. We prove that even with full knowledge of the future of the game, it is NP -complete to decide whether a given score is achievable at Kingdomino TM. [ABSTRACT FROM AUTHOR]

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B -polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B -polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B -polynomial, and try to answer the following questions: • what properties of the digraph can be detected from its B -polynomial (acyclicity, length of directed paths, number of strongly connected components, etc.)? • which of the marvelous properties of the Tutte polynomial carry over to the directed graph setting? The B -polynomial generalizes the strict chromatic polynomial of mixed graphs introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function version of the B -polynomial which simultaneously generalizes the Tutte symmetric function of Stanley and the quasisymmetric chromatic function of Shareshian and Wachs. [ABSTRACT FROM AUTHOR]

A strong edge-colouring of an undirected graph G is an edge-colouring where every two edges at distance at most2 receive distinct colours. The strong chromatic index of G is the least number of colours in a strong edge-colouring of G. A conjecture of Erdös and Nešetřil, stated back in the 80's, asserts that every graph with maximum degree Δ should have strong chromatic index at most roughly 1.25Δ2. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly 4Δ, and even to smaller values under additional structural requirements. In this work, we initiate the study of the strong chromatic index of 1-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of 1-planar graphs with maximum degree~Δ and strong chromatic index roughly 6Δ. As an upper bound, we prove that the strong chromatic index of a 1-planar graph with maximum degree Δ is at most roughly 24Δ (thus linear in Δ). The proof of this result is based on the existence of light edges in 1-planar graphs with minimum degree at least 3 [ABSTRACT FROM AUTHOR]

In the eternal domination game, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices and no more than one guard may occupy a vertex. The goal is to determine the eternal domination number of a graph which is the minimum number of guards required to defend the graph against an infinite sequence of attacks. In this paper, we continue the study of the eternal domination game on strong grids. Cartesian grids have been vastly studied with tight bounds for small grids such as 2×n, 3×n, 4×n, and 5×n grids, and recently it was proven in [Lamprou et al., CIAC 2017, 393-404] that the eternal domination number of these grids in general is within O(m+n) of their domination number which lower bounds the eternal domination number. Recently, Finbow et al. proved that the eternal domination number of strong grids is upper bounded by mn6+O(m+n). We adapt the techniques of [Lamprou et al., CIAC 2017, 393-404] to prove that the eternal domination number of strong grids is upper bounded by mn7+O(m+n). While this does not improve upon a recently announced bound of ⌈m3⌉⌈n3⌉+O(mn--√) [Mc Inerney, Nisse, Pérennes, CIAC 2019] in the general case, we show that our bound is an improvement in the case where the smaller of the two dimensions is at most 6179. [ABSTRACT FROM AUTHOR]

Trees are generalized to a special kind of higher dimensional complexes known as \((j,k)\)-trees ([L. W. Beineke, R. E. Pippert, On the structure of \((m,n)\)-trees, Proc. 8th S-E Conf. Combinatorics, Graph Theory and Computing, 1977, 75-80]), and which are a natural extension of \(k\)-trees for \(j=k-1\). The aim of this paper is to study\((k-r,k)\)-trees ([H. P. Patil, Studies on \(k\)-trees and some related topics, PhD Thesis, University of Warsaw, Poland, 1984]), which are a generalization of \(k\)-trees (or usual trees when \(k=1\)). We obtain the chromatic polynomial of \((k-r,k)\)-trees and show that any two \((k-r,k)\)-trees of the same order are chromatically equivalent. However, if \(r\neq 1\) in any \((k-r,k)\)-tree \(G\), then it is shown that there exists another chromatically equivalent graph \(H\), which is not a \((k-r,k)\)-tree. Further, the vertex-partition number and generalized total colourings of \((k-r,k)\)-trees are obtained. We formulate a conjecture about the chromatic index of \((k-r,k)\)-trees, and verify this conjecture in a number of cases. Finally, we obtain a result of [M. Borowiecki, W. Chojnacki, Chromatic index of \(k\)-trees, Discuss. Math. 9 (1988), 55-58] as a corollary in which \(k\)-trees of Class 2 are characterized.

Let m , n , λ be positive integers. What is the maximum number of proper vertex colorings in (at most) λ colors a graph with n vertices and m edges can have? On which graphs is this maximum attained? The question can be rephrased as the one of maximizing χ (G , λ) , the value of the chromatic polynomial of G at λ , over all graphs G with n vertices and m edges. This problem was stated independently by Wilf and Linial, and is still unsolved. In this article we survey the current state of the research directed at solving the problem. [ABSTRACT FROM AUTHOR]

We construct a new polynomial invariant for signed graphs, the trivari- ate Tutte polynomial, which contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte poly- nomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. While the Tutte polynomial of a graph is equivalently defined as the dichromatic polynomial or Whitney rank polynomial, the dichromatic polynomial of a signed graph (defined more widely for biased graphs by Zaslavsky) does not, by contrast, give the number of nowhere-zero flows as an evaluation in general. The trivariate Tutte polynomial contains Zaslavsky's dichromatic polynomial as a spe- cialization. Furthermore, the trivariate Tutte polynomial gives as an evaluation the number of proper colorings of a signed graph under a more general sense of signed graph coloring in which colors are elements of an arbitrary finite set equipped with an involution. [ABSTRACT FROM AUTHOR]

Let G be a threshold graph. In this paper, we give, in first hand, a formula relating the chromatic polynomial of G (the complement of G) to the chromatic polynomial of G: In second hand, we express the chromatic polynomials of G and G in terms of the generalized Bell polynomials. [ABSTRACT FROM AUTHOR]

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on α-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained. [ABSTRACT FROM AUTHOR]

Let S⊆V . A vertex uϵ V is a dominator of S if v dominates every vertex in S and v is said to be an anti-dominator of S if v dominates none of the vertices of S. Let C = (V1, V2, …, Vk) be a coloring of G and let v ϵ V (G). A color class Vi is called a dom-color class or an anti dom- color class of the vertex v according as v is a dominator of Vi or an anti- dominator of Vi. The coloring C is called a global dominator coloring of G if every vertex of G has a dom-color class and an anti dom-color class in C. The minimum number of colors required for a global dominator coloring of G is called the global dominator chromatic number and is denoted by Xgd(G). This paper initiates a study on this notion of global dominator coloring. [ABSTRACT FROM AUTHOR]

Abstract We study a family of polynomials associated with ascent–descent statistics on labeled rooted plane k -ary trees introduced by Gessel, from a rook-theoretic perspective. We generalize the excedance statistic on permutations to maximal nonattacking rook placements on certain rectangular boards by decomposing them into staircase boards. We then relate the number of maximal nonattacking rook placements on certain skew boards to the number of regions in extended Linial arrangements by establishing a relation between the factorial polynomial of those boards to the characteristic polynomial of extended Linial arrangements. Furthermore, we give a combinatorial interpretation of the number of bounded regions in extended Linial arrangements in terms of certain labeled rooted plane k -ary trees. Finally, using the work of Goldman–Joichi–White, we identify graphs whose chromatic polynomials equal the characteristic polynomials of extended Linial arrangements up to a straightforward normalization. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we focus on a special chromatic invariant of graphs, so-called the locating chromatic number. We give a sharp upper bound of the locating chromatic number of trees by using the number of leaves. [ABSTRACT FROM AUTHOR]

Abstract For an undirected graph G , the chromatic polynomial is a key specialisation of its Tutte polynomial. The " (α + n) -conjecture" states that given any algebraic integer α , for large enough n ∈ N the shifted value α + n is the root of some chromatic polynomial. We develop an algorithm whose input is a monic polynomial P d (X) ∈ Z [ X ] of degree d ≤ 4 , and the output is a (d , N) -biclique whose chromatic polynomial is a product of linear factors and the factor P d (X − n) where n ∈ N is taken to be sufficiently large, provided that such a (d , N) -biclique exists. In degree d = 3 our method tends to produce smaller (3 , N) -bicliques than Adam Bohn's cubic parameterisation (Bohn, 2014), while in degree d = 4 it provides an efficient way to check the (α + n) -conjecture for a given quartic polynomial. The construction relies on previous work of Farrell (1980) which interprets chromatic coefficients via induced subgraphs, together with a simple search algorithm that is based on the number of ways to partition an integer. We also derive new expressions for the interesting factors of the chromatic polynomials for (3 , N) -bicliques and (4 , N) -bicliques, in terms of these induced subgraphs. [ABSTRACT FROM AUTHOR]