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A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em fork } is a graph obtained from $K_{1,3}$ by subdividing an edge once, and an {\em odd torch} is a graph obtained from an odd hole by adding an edge $xy$ such that $x$ is non-adjacent to any vertex on the odd hole and the set of neighbors of $y$ on the odd hole is a stable set. Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54-60] proved that every (odd hole, bull)-free graph and every ($P_5$, bull)-free graph are perfectly divisible. Karthick {\em et al.} [The Electron. J. of Combin. 29 (2022) P3.19.] proved that every (fork, bull)-free graph is perfectly divisible. Chen and Xu [Discrete Appl. Math. 372 (2025) 298-307.] proved that every ($P_7,C_5$, bull)-free graph is perfectly divisible. Let $H\in$\{\{odd~torch\}, $\{P_8,C_5\}\}$. In this paper, we prove that every ($H$, bull)-free graph is perfectly divisible. We also prove that a ($P_6$, bull)-free graph is perfectly divisible if and only if it contains no Myci

The $3$-colorability problem is a well-known NP-complete problem and it remains NP-complete for $bull$-free graphs, where a $bull$ is the graph consisting of a $K_3$ with two pendant edges attached to two of its vertices. In this paper, for $k\geq3$, we characterize all $k$-colorable $(bull,claw)$-free graphs containing an induced cycle of length at least $6$. Moreover, we present the full characterization of all non $4$-colorable connected $(bull,claw)$-free graphs and $(bull,chair, C_5)$-free graphs, and all non $5$-colorable connected $(bull, claw, C_5)$-free graphs.

A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges. Hoàng [Discrete Math. 349 (2026) 114809] proposed four conjectures: 1. $P_5$-free graphs are perfectly divisible; 2. Odd hole-free graphs are perfectly divisible; 3. Even hole-free graphs are perfectly divisible; and 4. $4K_1$-free graphs are perfectly divisible. Karthick et al. [Electron. J. Combin. 29 (2022) P3.19] proposed a conjecture: Fork-free graphs are perfectly divisible. In this paper, we prove that all of five conjectures above hold for bull-free graphs. Our results also generalize some results of Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54--60] and Karthick et al. [Electron. J. Combin. 29 (2022) P3.19]. We say that a class ${\cal C}$ is {\em perfect-Pollyanna} if ${\cal C}\cap {\cal G}$ is perfectly divisible for any hereditary class ${\cal G}$ in which each triangle-free graph is 3-colorable. Let $H\in\{\text{house, hammer, diamond}\}$. In this paper, we prove that the class of $(\text{bull}, H)$-free gr

Financial markets exhibit highly dynamic and complex behaviors shaped by both historical price trajectories and exogenous narratives, such as news, policy interpretations, and social media sentiment. The heterogeneity in these data and the diverse insight of investors introduce biases that complicate the modeling of market dynamics. Unlike prior work, this paper explores the potential of bull and bear regimes in investor-driven market dynamics. Through empirical analysis on real-world financial datasets, we uncover a dynamic relationship between bias variation and behavioral adaptation, which enhances trend prediction under evolving market conditions. To model this mechanism, we propose the Bias to Behavior from Bull-Bear Dynamics model (B4), a unified framework that jointly embeds temporal price sequences and external contextual signals into a shared latent space where opposing bull and bear forces naturally emerge, forming the foundation for bias representation. Within this space, an inertial pairing module pairs temporally adjacent samples to preserve momentum, while the dual competition mechanism contrasts bullish and bearish embeddings to capture behavioral divergence. Togethe

In this paper, we are interested in $4$-colouring algorithms for graphs that do not contain an induced path on $6$ vertices nor an induced bull, i.e., the graph with vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and edge set $\{v_1v_2,v_2v_3,v_3v_4,v_2v_5,v_3v_5\}$. Such graphs are referred to as $(P_6,\text{bull})$-free graphs. A graph $G$ is \emph{$k$-vertex-critical} if $χ(G)=k$, and every proper induced subgraph $H$ of $G$ has $χ(H)<k$. In the current paper, we investigate the structure of $5$-vertex-critical $(P_6,\text{bull})$-free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial-time algorithm to decide if a $(P_6,\text{bull})$-free graph is $4$-colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4-colourability or non-4-colourability.

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb N$, if $G$ is a bull-free graph of clique number at most $k$ and every triangle-free induced subgraph of $G$ has chromatic number at most $t$, then $G$ has chromatic number at most $k^{O(\log t)}$. We further show that the bound $k^{O(\log t)}$ is best possible up to a multiplicative constant in the exponent. Thomassé, Trotignon, and Vušković (2017) were the first to give a bound of the form $2^{p\log p}$, where $p=O(k^2+t)$, with a proof that uses Chudnovsky's structure theorem for bull-free graphs. This was improved by Chudnovsky, Cook, Davies, and Oum (2026) to a bound of the form $k^{O(t)}$, with a 10-page proof that again relies heavily on Chudnovsky's structure theorem. Our proof is a single page long and completely avoids the structure theorem, instead using only a result of Chudnovsky and Safra (which itself has a short proof).

We study the Max Partial $k$-Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper $k$-coloring. For $k=1$ this problem coincides with Maximum Weight Independent Set, and for $k=2$ the problem is equivalent (by complementation) to Minimum Odd Cycle Transversal. Furthermore, it generalizes $k$-Coloring. We show that Max Partial $k$-Coloring on $n$-vertex instances with clique number $ω$ can be solved in time * $n^{\mathcal{O}(kω)}$ if the input graph excludes the bull and the chair as an induced subgraph, * $n^{\mathcal{O}(kω\log n)}$ if the input graph excludes the bull and E as an induced subgraph. This implies that $k$-Coloring can be solved in polynomial time in the former class, and in quasipolynomial-time in the latter one.

A graph $G$ has a perfect division if its vertex set can be partitioned into two sets $A$, $B$ such that $G[A]$ is perfect and $ω(G[B]) < ω(G)$. We call $G$ perfectly divisible if every induced subgraph of $G$ admits a perfect division. We prove that every ($P_2 \cup P_4$, bull)-free graph $G$ with $ω(G) \geq 3$ has a perfect division if $G$ contains no homogeneous set. The clique-number condition is tight: a counterexample exists for $ω(G) = 2$. Additionally, we present a short proof of the perfect divisibility of ($P_5$, bull)-free graphs, originally established by Chudnovsky and Sivaraman [J. Graph Theory 90 (2019), 54-60.].

The $3$-colourability problem is a well-known NP-complete problem and it remains NP-complete for $bull$-free graphs, where $bull$ is the graph consisting of $K_3$ with two pendant edges attached to two of its vertices. In this paper we study $3$-colourability of $(bull,H)$-free graphs for several graphs $H$. We show that these graphs are $3$-colourable or contain an induced odd wheel $W_{2p+1}$ for some $p\geq 2$ or a spindle graph $M_{3p+1}$ for some $p\geq 1$. Moreover, for all our results we can provide certifying algorithms that run in polynomial time.

Cryptocurrencies are highly volatile financial instruments with more and more new retail investors joining the scene with each passing day. Bitcoin has always proved to determine in which way the rest of the cryptocurrency market is headed towards. As of today Bitcoin has a market dominance of close to 50 percent. Bull and bear phases in cryptocurrencies are determined based on the performance of Bitcoin over the 50 Day and 200 Day Moving Averages. The aim of this paper is to foretell the performance of bitcoin in the near future by employing predictive algorithms. This predicted data will then be used to calculate the 50 Day and 200 Day Moving Averages and subsequently plotted to establish the potential bull and bear phases.

This paper discusses the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a partitionable (bull, diamond)-free graph can be found in polynomial time. Additionally, we show that the cop number of a (bull, diamond)-free graph containing a triangle is at most two less than its diameter. Furthermore, the cop number of a connected ($P_n$, bull, diamond)-free graph with a triangle, is at most $n-3$, for any natural number $n>3$. We also discuss a couple of applications of the structural theorem of the (bull, diamond)-free graphs in the conclusions.

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size $k$, when $k$ is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size $k$. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size $O(k^5)$. As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem of bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_t$ be the path and the cycle on $t$ vertices, respectively. A bull is the graph obtained from a triangle with two disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical ($P_5$,bull)-free graphs.

Very recently, Thomassé, Trotignon and Vuskovic [WG 2014] have given an FPT algorithm for Weighted Independent Set in bull-free graphs parameterized by the weight of the solution, running in time $2^{O(k^5)} \cdot n^9$. In this article we improve this running time to $2^{O(k^2)} \cdot n^7$. As a byproduct, we also improve the previous Turing-kernel for this problem from $O(k^5)$ to $O(k^2)$. Furthermore, for the subclass of bull-free graphs without holes of length at most $2p-1$ for $p \geq 3$, we speed up the running time to $2^{O(k \cdot k^{\frac{1}{p-1}})} \cdot n^7$. As $p$ grows, this running time is asymptotically tight in terms of $k$, since we prove that for each integer $p \geq 3$, Weighted Independent Set cannot be solved in time $2^{o(k)} \cdot n^{O(1)}$ in the class of $\{bull,C_4,\ldots,C_{2p-1}\}$-free graphs unless the ETH fails.

The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well as for odd-hole- and bull-free graphs (dart and bull have five vertices, say $a,b,c,d,e$, and dart has edges $ab,ac,ad,bd,cd,de$, while bull has edges $ab,bc,cd,be,ce$). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained.

A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph.

This paper examines the dynamic interaction between falling and rising markets for both the real and the financial sectors of the largest economy in the world using asymmetric causality tests. These tests require that each underlying variable in the model be transformed into partial sums of the positive and negative components. The positive components represent the rising markets and the negative components embody the falling markets. The sample period covers some part of the COVID19 pandemic. Since the data is non normal and the volatility is time varying, the bootstrap simulations with leverage adjustments are used in order to create reliable critical values when causality tests are conducted. The results of the asymmetric causality tests disclose that the bear markets are causing the recessions as well as the bull markets are causing the economic expansions. The causal effect of bull markets on economic expansions is higher compared to the causal effect of bear markets on economic recessions. In addition, it is found that economic expansions cause bull markets but recessions do not cause bear markets. Thus, the policies that remedy the falling financial markets can also help the

We give a polynomial-time algorithm that computes the chromatic number of any graph that contains no path on five vertices and no bull as an induced subgraph (where the bull is the graph with five vertices $a,b,c,d,e$ and edges $ab,bc,cd,be,ce$).

We give an $O(n^4)$ algorithm to find a minimum clique cover of a (bull, $C_4$)-free graph, or equivalently, a minimum colouring of a (bull, $2K_2$)-free graph, where $n$ is the number of vertices of the graphs.