Between proper and strong edge-colorings of subcubic graphs

In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced matching. We consider intermediate types of edge-colorings, where edges of some colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75) asserting that by allowing three additional induced matchings, one is able to save one matching color. We prove that every graph with maximum degree 3 can be decomposed into one matching and at most 8 induced matchings, and two matchings and at most 5 induced matchings. We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the above-mentioned conjecture for class I graphs

Further Evidence Towards the Multiplicative 1-2-3 Conjecture

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{\'c} can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes

Parameterized complexity of edge-coloured and signed graph homomorphism problems

We study the complexity of graph modification problems for homomorphism-based properties of edge-coloured graphs. A homomorphism from an edge-coloured graph $G$ to an edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. Given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph has a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are $2$-edge-coloured graphs whose colours are $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VERTEX DELETION $H$-COLOURING, EDGE DELETION $H$-COLOURING and SWITCHING $H$-COLOURING. These generalise $H$-COLOURING (where one has to decide if an input graph admits a homomorphism to $H$). Our main focus is when $H$ has order at most $2$, a case that includes standard problems such as VERTEX COVER, ODD CYCLE TRANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-complete complexity dichotomy for all three studied problems. Then, we address their parameterized complexity. We show that all VERTEX DELETION $H$-COLOURING and EDGE DELETION $H$-COLOURING problems for such $H$ are FPT. This is in contrast with the fact that already for some $H$ of order~$3$, unless P=NP, none of the three considered problems is in XP. We show that the situation is different for SWITCHING $H$-COLOURING: there are three $2$-edge-coloured graphs $H$ of order $2$ for which this is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$ for inputs of size $n$. For the other cases, SWITCHING $H$-COLOURING is FPT.Comment: 18 pages, 8 figures, 1 table. To appear in proceedings of IPEC 201

Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems. Our main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT

Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph $H$. The question we are interested in is: given an edge-coloured graph $G$, can we perform $k$ graph operations so that the resulting graph admits a homomorphism to $H$? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and $-$. We denote the corresponding problems (parameterized by $k$) by VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to a fixed target $H$). Our main focus is when $H$ is an edge-coloured graph with at most two vertices, a case that is already interesting as it includes problems such as VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph $H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$ are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the three problems is in XP, since 3-COLOURING is NP-c. We show that SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.Comment: 17 pages, 9 figures, 2 table

Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs

We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure

Further Evidence Towards the Multiplicative 1-2-3 Conjecture

The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kaziów in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vučković can be adapted to prove results on the product version. We prove that 4-chromatic graphs verify the product version of the 1-2-3 Conjecture. We also prove that for all graphs we can design 3-labellings that almost have the desired property. This leads to a new problem, that we solve for some graph classes

On various graph coloring problems

Dans cette thèse, nous étudions des problèmes de coloration de graphe. Nous nous intéressons à deux familles de colorations.La première consiste à colorer des graphes, appelés graphes signés, modélisant des relations sociales. Ceux-ci disposent de deux types d’arêtes : les arêtes positives pour représenter l’amitié et les arêtes négatives pour l’animosité. Nous pouvons colorer des graphes signés à travers la notion d’homomorphisme : le nombre chromatique d’un graphe signé (G, σ) est alors le nombre minimum de sommets d’un graphe signé (H, π) tel que (G, σ) admet un homomorphisme vers (H, π). Nous étudions la complexité des homomorphismes de graphes signés quand la cible est fixée et quand l’entrée peut être modifiée, et obtenons des dichotomies P/NP-complet et FPT/W[1]-difficile. Nous obtenons des bornes supérieures sur le nombre chromatique d’un graphe signé quand le graphe a peu de cycles. Enfin, nous étudions les relations entre les homomorphismes de graphes signés et le produit Cartésien des graphes signés.La deuxième famille de coloration consiste à colorer les arêtes au lieu des sommets en respectant différents critères. Nous étudions quatre types de colorations d’arêtes : la coloration d’arêtes « packing », la coloration d’arêtes injective, la coloration AVD et les 1-2-3-étiquetages. La coloration d’arêtes « packing » est une forme de coloration propre d’arêtes où chaque couleur a ses propres règles de conflits, par exemple, la couleur 1 pourrait obéir aux règles de la coloration propre d’arêtes tandis que la couleur 2 obéirait aux règles de la coloration forte d’arêtes. Nous étudions cette forme de coloration sur les graphes subcubiques en donnant des bornes supérieures sur le nombre de couleurs nécessaires pour colorer ces graphes. Une coloration d’arêtes injective est une coloration d’arêtes telle que pour chaque chemin de longueur 3, les deux arêtes aux extrémités du chemin n’ont pas la même couleur. Nous déterminons la complexité de la coloration d’arêtes injective sur plusieurs classes de graphes. Pour les colorations AVD, c’est-à-dire les colorations propres d’arêtes où les sommets adjacents sont incidents à des ensembles de couleurs différents, nous obtenons des bornes supérieures sur le nombre de couleurs requises pour colorer le graphe quand le degré maximum du graphe est significativement plus grand que son degré moyen maximum, ou quand le graphe est planaire et a un degré maximum supérieur ou égal à 12. Finalement, nous prouvons la 1-2-3 Conjecture multiplicative : pour tout graphe connexe (non réduit à une arête), on peut colorer ses arêtes avec les couleurs 1, 2 et 3 de telle manière que la coloration (de sommets) obtenue en associant à un sommet le produit des couleurs de ses arêtes incidentes est propre.In this thesis, we study some graph coloring problems. We are interested intwo families of colorings.The first one consists in coloring graphs, called signed graphs, modeling social links. These signed graphs dispose of two types of edges: positive edges to represent friendship and negative edges for animosity. Coloring signed graphs is done through the notion of homomorphism: the chromatic number of a signed graph (G, σ) is the smallest order of a signed graph (H, π) to which (G, σ) admits a homomorphism. We study the complexity of homomorphisms of signed graphs when the target graph is fixed and when the input can be modified, giving P/NP-complete dichotomies and FPT/W[1]-hard dichotomies. We also present bounds on the chromatic number of signed graphs when the input graph has few cycles. Finally, we study the relationship between homomorphisms of signed graphs and the Cartesian product of signed graphs.The second family of colorings consists in coloring edges, instead of vertices, according to some constraints. We study four kinds of edge-colorings notions: packing edge-colorings, injective edge-colorings, AVD colorings and 1-2-3-labellings. Packing edge-coloring is a form of proper edge-coloring where each color has its own conflict rule, for example, color 1 may behave according to the rules of proper edge-colorings while color 2 behave according to the rules of strong edge-colorings. We study packing edge-coloring on subcubic graphs and provide bounds on the number of colors necessary to color the graphs. An injective edge-coloring is an edge-coloring where for any path of length 3, the two non-internal edges of the path receive different colors. We determine the complexity of injective edge-coloring for some classes of graphs. For AVD colorings, i.e. a proper edge-coloring where adjacent vertices are incident with different sets of colors, we obtain bounds on the number of colors required to color the graph when the graph has its maximum degree significantly greater than its maximum average degree and when the graph is planar and has maximum degree at least 12. Finally, we prove the Multiplicative 1-2-3 Conjecture, i.e. that every connected graph (which is not just an edge) can be edge-labelled with labels 1, 2 and 3 so that the coloring of G, obtained by associating with each vertex the product of the labels on edges incident with u, is proper

Sur divers problèmes de coloration de graphes

Dans cette thèse, nous étudions des problèmes de coloration de graphe. Nous nous intéressons à deux familles de colorations.La première consiste à colorer des graphes, appelés graphes signés, modélisant des relations sociales. Ceux-ci disposent de deux types d’arêtes : les arêtes positives pour représenter l’amitié et les arêtes négatives pour l’animosité. Nous pouvons colorer des graphes signés à travers la notion d’homomorphisme : le nombre chromatique d’un graphe signé (G, σ) est alors le nombre minimum de sommets d’un graphe signé (H, π) tel que (G, σ) admet un homomorphisme vers (H, π). Nous étudions la complexité des homomorphismes de graphes signés quand la cible est fixée et quand l’entrée peut être modifiée, et obtenons des dichotomies P/NP-complet et FPT/W[1]-difficile. Nous obtenons des bornes supérieures sur le nombre chromatique d’un graphe signé quand le graphe a peu de cycles. Enfin, nous étudions les relations entre les homomorphismes de graphes signés et le produit Cartésien des graphes signés.La deuxième famille de coloration consiste à colorer les arêtes au lieu des sommets en respectant différents critères. Nous étudions quatre types de colorations d’arêtes : la coloration d’arêtes « packing », la coloration d’arêtes injective, la coloration AVD et les 1-2-3-étiquetages. La coloration d’arêtes « packing » est une forme de coloration propre d’arêtes où chaque couleur a ses propres règles de conflits, par exemple, la couleur 1 pourrait obéir aux règles de la coloration propre d’arêtes tandis que la couleur 2 obéirait aux règles de la coloration forte d’arêtes. Nous étudions cette forme de coloration sur les graphes subcubiques en donnant des bornes supérieures sur le nombre de couleurs nécessaires pour colorer ces graphes. Une coloration d’arêtes injective est une coloration d’arêtes telle que pour chaque chemin de longueur 3, les deux arêtes aux extrémités du chemin n’ont pas la même couleur. Nous déterminons la complexité de la coloration d’arêtes injective sur plusieurs classes de graphes. Pour les colorations AVD, c’est-à-dire les colorations propres d’arêtes où les sommets adjacents sont incidents à des ensembles de couleurs différents, nous obtenons des bornes supérieures sur le nombre de couleurs requises pour colorer le graphe quand le degré maximum du graphe est significativement plus grand que son degré moyen maximum, ou quand le graphe est planaire et a un degré maximum supérieur ou égal à 12. Finalement, nous prouvons la 1-2-3 Conjecture multiplicative : pour tout graphe connexe (non réduit à une arête), on peut colorer ses arêtes avec les couleurs 1, 2 et 3 de telle manière que la coloration (de sommets) obtenue en associant à un sommet le produit des couleurs de ses arêtes incidentes est propre.In this thesis, we study some graph coloring problems. We are interested intwo families of colorings.The first one consists in coloring graphs, called signed graphs, modeling social links. These signed graphs dispose of two types of edges: positive edges to represent friendship and negative edges for animosity. Coloring signed graphs is done through the notion of homomorphism: the chromatic number of a signed graph (G, σ) is the smallest order of a signed graph (H, π) to which (G, σ) admits a homomorphism. We study the complexity of homomorphisms of signed graphs when the target graph is fixed and when the input can be modified, giving P/NP-complete dichotomies and FPT/W[1]-hard dichotomies. We also present bounds on the chromatic number of signed graphs when the input graph has few cycles. Finally, we study the relationship between homomorphisms of signed graphs and the Cartesian product of signed graphs.The second family of colorings consists in coloring edges, instead of vertices, according to some constraints. We study four kinds of edge-colorings notions: packing edge-colorings, injective edge-colorings, AVD colorings and 1-2-3-labellings. Packing edge-coloring is a form of proper edge-coloring where each color has its own conflict rule, for example, color 1 may behave according to the rules of proper edge-colorings while color 2 behave according to the rules of strong edge-colorings. We study packing edge-coloring on subcubic graphs and provide bounds on the number of colors necessary to color the graphs. An injective edge-coloring is an edge-coloring where for any path of length 3, the two non-internal edges of the path receive different colors. We determine the complexity of injective edge-coloring for some classes of graphs. For AVD colorings, i.e. a proper edge-coloring where adjacent vertices are incident with different sets of colors, we obtain bounds on the number of colors required to color the graph when the graph has its maximum degree significantly greater than its maximum average degree and when the graph is planar and has maximum degree at least 12. Finally, we prove the Multiplicative 1-2-3 Conjecture, i.e. that every connected graph (which is not just an edge) can be edge-labelled with labels 1, 2 and 3 so that the coloring of G, obtained by associating with each vertex the product of the labels on edges incident with u, is proper

Sur divers problèmes de coloration de graphes

In this thesis, we study some graph coloring problems. We are interested intwo families of colorings.The first one consists in coloring graphs, called signed graphs, modeling social links. These signed graphs dispose of two types of edges: positive edges to represent friendship and negative edges for animosity. Coloring signed graphs is done through the notion of homomorphism: the chromatic number of a signed graph (G, σ) is the smallest order of a signed graph (H, π) to which (G, σ) admits a homomorphism. We study the complexity of homomorphisms of signed graphs when the target graph is fixed and when the input can be modified, giving P/NP-complete dichotomies and FPT/W[1]-hard dichotomies. We also present bounds on the chromatic number of signed graphs when the input graph has few cycles. Finally, we study the relationship between homomorphisms of signed graphs and the Cartesian product of signed graphs.The second family of colorings consists in coloring edges, instead of vertices, according to some constraints. We study four kinds of edge-colorings notions: packing edge-colorings, injective edge-colorings, AVD colorings and 1-2-3-labellings. Packing edge-coloring is a form of proper edge-coloring where each color has its own conflict rule, for example, color 1 may behave according to the rules of proper edge-colorings while color 2 behave according to the rules of strong edge-colorings. We study packing edge-coloring on subcubic graphs and provide bounds on the number of colors necessary to color the graphs. An injective edge-coloring is an edge-coloring where for any path of length 3, the two non-internal edges of the path receive different colors. We determine the complexity of injective edge-coloring for some classes of graphs. For AVD colorings, i.e. a proper edge-coloring where adjacent vertices are incident with different sets of colors, we obtain bounds on the number of colors required to color the graph when the graph has its maximum degree significantly greater than its maximum average degree and when the graph is planar and has maximum degree at least 12. Finally, we prove the Multiplicative 1-2-3 Conjecture, i.e. that every connected graph (which is not just an edge) can be edge-labelled with labels 1, 2 and 3 so that the coloring of G, obtained by associating with each vertex the product of the labels on edges incident with u, is proper.Dans cette thèse, nous étudions des problèmes de coloration de graphe. Nous nous intéressons à deux familles de colorations.La première consiste à colorer des graphes, appelés graphes signés, modélisant des relations sociales. Ceux-ci disposent de deux types d’arêtes : les arêtes positives pour représenter l’amitié et les arêtes négatives pour l’animosité. Nous pouvons colorer des graphes signés à travers la notion d’homomorphisme : le nombre chromatique d’un graphe signé (G, σ) est alors le nombre minimum de sommets d’un graphe signé (H, π) tel que (G, σ) admet un homomorphisme vers (H, π). Nous étudions la complexité des homomorphismes de graphes signés quand la cible est fixée et quand l’entrée peut être modifiée, et obtenons des dichotomies P/NP-complet et FPT/W[1]-difficile. Nous obtenons des bornes supérieures sur le nombre chromatique d’un graphe signé quand le graphe a peu de cycles. Enfin, nous étudions les relations entre les homomorphismes de graphes signés et le produit Cartésien des graphes signés.La deuxième famille de coloration consiste à colorer les arêtes au lieu des sommets en respectant différents critères. Nous étudions quatre types de colorations d’arêtes : la coloration d’arêtes « packing », la coloration d’arêtes injective, la coloration AVD et les 1-2-3-étiquetages. La coloration d’arêtes « packing » est une forme de coloration propre d’arêtes où chaque couleur a ses propres règles de conflits, par exemple, la couleur 1 pourrait obéir aux règles de la coloration propre d’arêtes tandis que la couleur 2 obéirait aux règles de la coloration forte d’arêtes. Nous étudions cette forme de coloration sur les graphes subcubiques en donnant des bornes supérieures sur le nombre de couleurs nécessaires pour colorer ces graphes. Une coloration d’arêtes injective est une coloration d’arêtes telle que pour chaque chemin de longueur 3, les deux arêtes aux extrémités du chemin n’ont pas la même couleur. Nous déterminons la complexité de la coloration d’arêtes injective sur plusieurs classes de graphes. Pour les colorations AVD, c’est-à-dire les colorations propres d’arêtes où les sommets adjacents sont incidents à des ensembles de couleurs différents, nous obtenons des bornes supérieures sur le nombre de couleurs requises pour colorer le graphe quand le degré maximum du graphe est significativement plus grand que son degré moyen maximum, ou quand le graphe est planaire et a un degré maximum supérieur ou égal à 12. Finalement, nous prouvons la 1-2-3 Conjecture multiplicative : pour tout graphe connexe (non réduit à une arête), on peut colorer ses arêtes avec les couleurs 1, 2 et 3 de telle manière que la coloration (de sommets) obtenue en associant à un sommet le produit des couleurs de ses arêtes incidentes est propre