backtrack.js

const getNextMoves = (x, y) => ([
    [x+3, y, '⇒'],
    [x, y-3, '⇑'],
    [x-3, y, '⇐'],
    [x, y+3, '⇓'],

    [x+2, y+2, '⇘'],
    [x-2, y-2, '⇖'],
    [x-2, y+2, '⇙'],
    [x+2, y-2, '⇗']
]);

const isValidMove = (previousMoves, x, y) => {
    if(previousMoves.includes(`${x},${y}`)) return false;

    return (x >= 0 && x <= 9) && (y >= 0 && y <= 9)
};

const solve = (x, y, previousMoves, directions = []) => {
    if(previousMoves.length === 100) return [previousMoves, directions];

    for(const [nextX, nextY, direction] of getNextMoves(x, y)) {
        if(!isValidMove(previousMoves, nextX, nextY)) {
            continue;
        }

        previousMoves.push(`${nextX},${nextY}`);
        directions.push(direction)
        let result = solve(nextX, nextY, previousMoves, directions);
        if(result) return result;
        previousMoves.pop();
        directions.pop()
    }

    return false;
}

const [movelist ,directions] = solve(0, 0, [`0,0`]);
console.log(directions.map((x, i) => `${i+1}:${x}`).join("\n"))
console.log({movelist})

driver.js

const list = [
    '0,0', '3,0', '6,0', '9,0', '9,3', '6,3', '3,3', '0,3',
    '0,6', '3,6', '6,6', '9,6', '9,9', '6,9', '3,9', '0,9',
    '2,7', '5,7', '8,7', '8,4', '8,1', '5,1', '2,1', '2,4',
    '5,4', '7,6', '7,3', '7,0', '4,0', '1,0', '1,3', '4,3',
    '4,6', '1,6', '1,9', '4,9', '7,9', '9,7', '9,4', '9,1',
    '6,1', '3,1', '0,1', '0,4', '3,4', '6,4', '6,7', '3,7',
    '0,7', '2,9', '5,9', '8,9', '8,6', '8,3', '8,0', '5,0',
    '2,0', '2,3', '5,3', '3,5', '6,5', '9,5', '9,8', '6,8',
    '3,8', '0,8', '0,5', '0,2', '3,2', '6,2', '9,2', '7,4',
    '7,1', '4,1', '1,1', '1,4', '1,7', '4,7', '7,7', '5,5',
    '5,2', '8,2', '8,5', '8,8', '5,8', '2,8', '2,5', '2,2',
    '4,4', '2,6', '5,6', '7,8', '7,5', '7,2', '4,2', '1,2',
    '1,5', '4,5', '4,8', '1,8'
]

const sleep = (ms) => new Promise((res) => setTimeout(res, ms));

const drive = async (movelist) => {
    for(const move of movelist){
        const [x,y] = move.split(",")
        document.getElementById(`${y}_${x}`).click();
        await sleep(150);
    }
}


drive(list).catch(console.error)

warnsdorffs.js

const GAME_BOARD_SIZE = process.env['BOARD_SIZE'] || 10;

const getNextMoves = (x, y) => ([
    [x+3, y],
    [x, y-3],
    [x-3, y],
    [x, y+3],

    [x+2, y+2],
    [x-2, y-2],
    [x-2, y+2],
    [x+2, y-2]
]);

const randomBetween = (min = 0, max = 0) => {
    return Math.floor(Math.random() * (max+1 - min) + min);
}

const randomXY = () => [randomBetween(0, GAME_BOARD_SIZE-1), randomBetween(0, GAME_BOARD_SIZE-1)];

const isValidMove = (x, y, board) => {
    if(board[y*GAME_BOARD_SIZE+x]) return false;

    return (x >= 0 && x <= GAME_BOARD_SIZE-1) && (y >= 0 && y <= GAME_BOARD_SIZE-1)
};

const getDegree = (x, y, board) => {
    let count = 0;

    for(const[nextX, nextY] of getNextMoves(x, y)){
        if(isValidMove(nextX, nextY, board)){
            count += 1;
        }
    }

    return count;
}

const solve = (x, y, previousMoves, board) => {
    previousMoves.push(`${x},${y}`);
    board[y*GAME_BOARD_SIZE+x] = 1;

    if(previousMoves.length === GAME_BOARD_SIZE*GAME_BOARD_SIZE){
        return previousMoves
    }

    const nextPositions = getNextMoves(x, y).filter(([nextX, nextY]) => isValidMove(nextX, nextY, board));
    const [minimumDegree] = nextPositions.sort(([xA, yA], [xB, yB]) => getDegree(xA, yA, board) - getDegree(xB, yB, board));
    const [nextX, nextY] = minimumDegree;

    return solve(nextX, nextY, previousMoves, board);
}

const [x, y] = randomXY();

const result = solve(x, y, [], Array(GAME_BOARD_SIZE*GAME_BOARD_SIZE).fill(0));

console.log(result)