The P-3-convex hull H(S) of a set S subset of V(G) is the smallest set containing S such that, for every pair of vertices u, v is an element of S, all vertices on any path of three vertices between them are also included in the set. The P-3-Carath & eacute;odory number c(G) is the smallest integer c such that, for every set S and every v E H(S), there exists a subset S ' c S of size at most c, such that v is an element of H(S '). In 2014, Barbosa et al. proved that, for a given graph G and a positive integer k, the problem of deciding whether c(G) >= k is NP-complete. In this paper, we present upper bounds on the P-3-Caratheodory number of graphs with diameter two, and establish that: c(G) = 2 for graphs with cut-vertex; c(G) < (5+root 8 triangle-15)/(2 )for biconnected graphs; and c(G) <= 4 for biconnected C6-free graphs. Moreover, we show a polynomial-time algorithm for biconnected diameter-two graphs, that constructs a special path from which we compute an upper bound for the Caratheodory number. These results also provide supporting evidence for the conjecture that the P-3-Caratheodory number is bounded by a constant in graphs with a limited diameter. (c) 2026 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

A graceful k-coloring of a graph G is a proper vertex coloring pi : V(G) -> {1, 2, ... , k}, k >= 1, that induces a proper edge coloring pi ': E(G) -> {1,2, ...,k - 1} defined by pi '(uv) = |pi(u) - pi(v)|, where uv is an element of E(G). The smallest positive integer k for which a graph G has a graceful k-coloring is called the graceful chromatic number of G and is denoted by chi g(G). It is well known that any tree T of maximum degree triangle has graceful chromatic number chi g(T) <= 5 3 triangle. In this paper, we refine this bound by showing that, under certain restrictions, a tree with maximum degree triangle has a graceful chromatic number of either triangle+1 or at most triangle+2. Additionally, we identify two families of lobsters with maximum degree 3: one with a graceful chromatic number of 4 and another with a graceful chromatic number of 5, showing that both lower and upper bounds for this parameter are attained within this class of graphs. Furthermore, we develop a polynomial-time algorithm to determine whether a given tree T with maximum degree three has a graceful chromatic number of 4. Finally, we present a proof that the GRACEFUL 4-COLORING PROBLEM is NP-complete. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Total-Neighbor-Distinguishing Index by Sums of a graph G ( TNDI_∑(G) ) is the smallest positive integer for which there exists a total coloring such that the sum of the color assigned to a vertex and the colors of all its incident edges is distinct, for every pair of adjacent vertices u and w. Pilsniak and Wozniak (2015) introduced this notion and conjectured that TNDI_∑ is less than or equal to the maximum degree plus 3. In 2011, Gravier et al. introduced the concept of generalized Sierpiński graphs, denoted S(n, G), which extends the notion of Sierpiński graphs given by Hinz et al. (2017) by replacing the complete graph in their definition with any graph G. In this paper, we verify that S(n, G) graphs satisfy the TNDI_∑ Conjecture, when G is a bipartite graph, a path, a cycle, a star or a complete graph. Furthermore, we determine the TNDI_∑(S(n,G)) when G is a complete bipartite, a path, a cycle, a star or a regular bipartite graph. Remarkably, these colorings also satisfy the corresponding conjecture for the Adjacent-Vertex-Distinguishing Total Coloring (AVDTC), and the TNDI_∑ of these graphs coincides with their AVDTC chromatic number. Moreover, we show that the TNDI_∑ Conjecture is satisfied for the graph classes of the regularizations +S_p^n and ++S_p^n of Sierpiński graphs S_p^n , which contributes to proving that the conjecture holds for their respective fractals as well.

The i-iterated subdivided-line graph Γi(G) of a base graph G is the graph obtained from G by iteratively applying the subdividedline graph operation i times. The M-polynomial of a graph G is a bivariate polynomial that encodes the degree-based properties of G. In this paper, we present a general formula expressing the Mpolynomial of Γi (G) in terms of the degrees of the vertices of the base graph G. We compute the First and Second Zagreb indices from the M-polynomial of i-iterated subdivided-line graphs Γi (G) when G belongs to several graph classes, such as wheel, ladder, ∆- regular, cycle, and tadpole graphs. The obtained results generalize those of Ranjini et al. (2011). Additionally, we analyse the impact of the vertex of maximum degree on the value of the First and Second Zagreb indices, providing asymptotic upper bounds for i-iterated subdivided-line graphs of general graphs.

The coloring game is a two-player non-cooperative game conceived in 1981. Alice and Bob alternate turns to properly color the vertices of a finite, simple, undirected, and connected graph G with t colors. Alice's goal is to properly color the vertices of G with t colors; Bob's aim is to prevent it. If, at any point, there is an uncolored vertex without an available color, Bob wins; otherwise, Alice wins. The game chromatic number chi g(G) is the smallest t for Alice to have a winning strategy. Motivated by the study of Obszarski et al. (2023) in complete multipartite graphs with no singleton parts, we contribute to the understanding of the difficulty of allowing singleton parts in the complete multipartite graph and obtain surprising results even if we allow just one singleton part. (c) 2026 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Snarks are a historical class of cubic graphs with peculiar properties motivated by the Four-Color Theorem. In nearly 100 years of search, since its definition by Peter Guthrie Tait in 1880, only five such graphs were identified which motivated Martin Gardner in 1976 to call them snark, a mysterious creature. In 1975, Rufus Isaacs introduced a method known as dot product, which allowed the construction of new snarks from known snarks, and presented the first infinite family of snarks. A new method proposed in 1996 by Martin Kochol allowed to obtaining new snarks from smaller graphs, known as Kochol superposition. However, this method was usually used to obtain snarks with large girth. We applied the Kochol superposition to known snarks: the family of Goldberg snarks (known to be Type 1) and a girth 4 snark recently discovered by Gunnar Brinkmann et al. (known to be Type 2). Surprisingly, when we apply the Kochol superposition to a Type 1 snark with a Type 2 snark, we can obtain new families of snarks of distinct Types: Type 1 and Type 2.

A graceful k -coloring of a graph G consists of a proper vertex coloring f : V ( G ) → { 1, …, k } that induces a proper edge coloring. In this case, the color assigned to an edge u v ∈ E ( G ) is determined by the absolute value of the difference between the colors assigned to vertices u and v. The minimum k for which a graph G has a graceful k-coloring is the graceful chromatic number of G, denoted by χ g ( G ). In this paper, we establish the first known upper bound for the graceful chromatic number of an arbitrary graph, in terms of its maximum degree Δ, i.e., we prove that the graceful chromatic number satisfies the inequality χ g ( G ) ≤ 5 Δ 2 − 3 Δ 2 + 1. This result provides the first upper bound for χ g ( K n ) of complete graphs K n, and represents an improvement over a previous finding by Bi et al. (2017) for regular complete k-partite graphs K k ( p ). We demonstrate the NP-completeness of the problem of determining whether a graph G has a graceful 5-coloring. Furthermore, we extend our investigation to subcubic graphs, establishing upper bounds for the graceful chromatic number of families of subcubic graphs without adjacent vertices of maximum degree. Additionally, we determine the graceful chromatic number for two cubic graph classes, namely Flower snarks and Goldberg snarks.

Snarks are cubic graphs with peculiar properties, making them relevant to several problems in graph theory, such as edge and total coloring. While infinite families of Type 1 snarks are well known, Type 2 snarks remain rare and difficult to construct. In 1996, Kochol introduced a method for constructing new snarks by combining two known snarks, usually Type 1. We apply Kochol’s superposition to obtain new Type 2 snarks. As a result, we construct an infinite family of Type 2 snarks with girth 4 by Kochol’s superposition of a known Type 2 snark with the infinite family of Goldberg snarks.

We investigate the graceful k-coloring introduced by Gary Chartrand in 2015. The graceful k-coloring of a graph G consists of a proper vertex coloring pi : V(G) -> {1, 2, ... , k}, k >= 1, that induces a proper edge coloring pi ': E(G) -> {1,2, ...,k - 1} defined by pi '(uv) = |pi(u) - pi(v)|, where uv is an element of E(G). The smallest positive integer k for which a graph G has a graceful k-coloring is called the graceful chromatic number of G and is denoted by chi g(G). In this paper, we improve a previous upper bound for the graceful chromatic number of an arbitrary graph, showing that chi g(G) <= 2A2 - A + 1, where A is the maximum degree of the graph G. This also implies an improvement over the first bound given for complete graphs. Moreover, we study this problem within the context of cubic graph classes, determining the exact value of the graceful chromatic number of each member of the infinite families of the Generalized Blanu & scaron;a and the Loupekine snarks. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In 2003, the frequency assignment problem in a cellular network motivated Even et al. to introduce a new coloring problem: Conflict-Free coloring. Inspired by this problem and by the Gardner-Bodlaender's coloring game, in 2020, Chimelli and Dantas introduced the Conflict-Free Closed Neighborhood \(k\)-coloring game (CFCN \(k\)-coloring game). The game starts with an uncolored graph \(G\), \(k\geq 2\) different colors, and two players, Alice and Bob, who alternately color the vertices of \(G\). Both players can start the game and respect the following legal coloring rule: for every vertex \(v\), if the closed neighborhood \(N[v]\) of \(v\) is fully colored then there exists a color that was used only once in \(N[v]\). Alice wins if she ends up with a Conflict-Free Closed Neighborhood \(k\)-coloring of \(G\), otherwise, Bob wins if he prevents it from happening. In this paper, we introduce the game for open neighborhoods, the Conflict-Free Open Neighborhood \(k\)-coloring game (CFON \(k\)-coloring game), and study both games on graph classes determining the least number of colors needed for Alice to win the game.

Uma k-coloração graciosa de um grafo G consiste em uma coloração própria de vértices f : V (G) → {1, 2, . . . , k}, k ≥ 1, que induz uma coloração própria de arestas f ′ : E(G) → {1, 2, . . . , k − 1} definida por f ′(uv) = |f(u)− f(v)|, onde u, v ∈ V (G). Colorações graciosas foram introduzidas por Gary Chartrand, por volta de 2015, como uma variação da conhecida rotulação graciosa introduzida por Alexander Rosa por volta de 1967. Numerosos artigos foram publicados sobre ambos os temas e vários problemas desafiadores permanecem em aberto. Neste trabalho, investigamos colorações graciosas no contexto de classes de grafos cúbicos como Snarks de Blanuša Generalizados Bi1, para i ≥ 1, e determinamos que o menor inteiro positivo k para o qual Bi1 possui uma k-coloração graciosa é 6.

The coloring game is a two-player non-cooperative game conceived in 1981. Alice and Bob alternate turns to properly color the vertices of a finite graph G with t colors. Alice’s goal is to properly color the vertices of G with t colors; Bob’s aim is to prevent it. If, at any point, there is an uncolored vertex without an available color, Bob wins; otherwise, Alice wins. The game chromatic number χg(G) is the smallest t for Alice to have a winning strategy. In 1991, Bodlaender showed that a caterpillar was the smallest tree T with χg(T) = 4; in 1993, Faigle et al. proved χg(T) ≤ 4 for every tree T. In 2015, Dunn et al. proposed the characterization of forests with game chromatic numbers 3 and 4. In this paper, we extend results from caterpillars to more general trees, and establish sufficient conditions to ensure that a tree has game chromatic number 4.

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that each edge is uniquely identified by the absolute difference of the labels of its endpoints. In this work, we study the graceful labeling problem in the context of maker-breaker graph games. The Graceful Game was introduced by Tuza, in 2017, as a two-players game on a connected graph in which the players, Alice and Bob, take moves labeling the vertices with distinct integers from 0 to m . Players are constrained to use only legal labelings (moves), that is, after a move, all edge labels are distinct. Alice’s goal is to obtain a graceful labeling for the graph, as Bob’s goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, trees, gear graphs, web graphs, prisms, hypercubes, 2-powers of paths, wheels and fan graphs.

The (a,b)-monochromatic transversal game is an avoider-enforcer combinatorial game in which two players, Alice and Bob, alternately take turns colouring respectively a vertices in red and b vertices in blue of a hypergraph. She wins the game by obtaining a red transversal while he wins by obtaining a monochromatic blue hyperedge. Also, both players are enabled to start the game and they play optimally. In this paper, we analyze the game played on biclique-hypergraphs of powers of paths and of powers of cycles showing strategies that, depending on the choice of the parameters a and b, allow a specific player to win the game.

The Edge-Sum Distinguishing game (ESD game) is a graph labeling game proposed by Tuza in 2017. In such a game, the players, traditionally called Alice and Bob, alternately assign an unused label f (v) is an element of {1, ... , s} to an unlabeled vertex v of a graph G, and the induced edge label phi(uv) of an edge uv is an element of E(G) is given by phi(uv) = f (u) + f (v). Alice's goal is to end up with an injective vertex labeling of all vertices of G that induces distinct edge labels, and Bob's goal is to prevent this. Tuza also posed the following questions about the ESD game: given a simple graph G, for which values of s can Alice win the ESD game? And if Alice wins the ESD game with the set of labels {1, ... , s}, can she also win with {1, ... , s + 1}? In this work, we partially answer these questions by presenting bounds on the number of consecutive non-negative integer labels necessary for Alice to win the ESD game on general and classical families of graphs.

Abstract Snarks are cyclically 4-edge-connected cubic graphs that admit no proper 3-edge-coloring. A snark is of Type 1 if it has a proper total coloring of its vertices and edges with four colors; it is of Type 2 if any total coloring requires at least five colors. Following an extensive computer search, Cavicchioli et al. (2003) asked whether there exist Type 2 snarks of girth at least 5. This question is still open, however, in 2015, Brinkmann et al. described the first known family of Type 2 snarks of girth 4. In this work we provide new families of Type 2 snarks of girth 4, all of which can be constructed by a dot product of two Type 1 snarks. We also show that the previously constructed Type 2 snarks of Brinkmann et al. do not have this property.

A fullerene graph is a planar, cubic, 3-connected graph with only pentagonal and hexagonal faces. In 2012, Andova and Skrekovski conjectured that the diameter of every fullerene graph with n vertices is at least left perpendicular root 5n/3 left perpendicular - 1. They computed this lower bound by studying a particular class of fullerene graphs named spherical with icosahedral symmetry. We denote these graphs by G(i,j), by setting two parameters i, j is an element of N*, such that i <= j. In their study, Andova and Skrekovski offered numerous properties of hexagonal lattices and calculated the diameter of two remarkable spherical fullerene graphs: G(0,j) and G(j,j). Although the conjecture is valid for these two distinct classes, it remains open deciding whether the premise is proper for all spherical fullerene graphs G(i,j). In this work, we present the first class of fullerene graphs with icosahedral symmetry that do not satisfy Andova and Skrekovski's conjecture, which refutes that this conjecture is valid for all spherical graphs. We also focus on showing properties of spherical fullerene graphs and the hexagonal lattice itself. We prove that all graphs G(i,j) have a reduction of the form G(i-k,j-k), where k <= i, such that their triangular faces are entirely contained in the triangular faces of G(i,j). In addition, by setting k = i, this property states a particular link among G(i,j), G(i-1, j-1), , G(0,j-i), creating a chain of reductions of G(i,j), which implies that diam (G(i,j)) >= diam (G(0,j-i)). (C) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)

The coloring game is played by Alice and Bob on a finite graph G. They take turns properly coloring the vertices with t colors. The goal of Alice is to color the input graph with t colors, and Bob does his best to prevent it. If at any point there exists an uncolored vertex without available color, then Bob wins; otherwise Alice wins. The game chromatic number χg(G) of G is the smallest number t such that Alice has a winning strategy. In 1991, Bodlaender showed the smallest tree T with χg(T) equal to 4, and in 1993 Faigle et al. proved that every tree T satisfies the upper bound χg(T)≤4. The stars T = K1,p with p ≥ 1 are the only trees satisfying χg(T) = 2; and the paths T = Pn, n ≥ 4, satisfy χg(T) = 3. Despite the vast literature in this area, there does not exist a characterization of trees with χg(T) = 3 or 4. We answer a question about the required degree to ensure χg(T) = 4, by exhibiting infinitely many trees with maximum degree 3 and game chromatic number 4.

The Adjacent-Vertex-Distinguishing Total Coloring (AVDTC) conjecture asserts that every simple graph can be colored with an AVDTC using at most Δ+3 colors. We show that this conjecture is satisfied for all the Sierpiński graphs Spn, their regularizations +Spn and ++Spn and the fractals obtained as limit space of these graphs. Moreover, we construct explicit total colorings (with at most Δ+2 colors) for Sierpiński triangles graphs STpn and Sierpiński triangle fractal with the limit space STp∞, for p∈{3,4,5,6}, and prove that the AVDTC conjecture is also valid for these cases.